Swarming Behaviors in Multiagent Systems with Nonlinear...
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Research Article Swarming Behaviors in Multiagent Systems with Nonlinear Dynamics and Aperiodically Intermittent Communication Anding Dai , 1,2 Yicheng Liu , 1 Dongyun Yi, 1 and Peiying Xiong 2 1 College of Science, National University of Defense Technology, Changsha 410073, China 2 School of Mathematics and Computer Science, Hunan City University, Yiyang 413000, China Correspondence should be addressed to Yicheng Liu; [email protected] Received 9 September 2018; Revised 1 December 2018; Accepted 6 January 2019; Published 23 January 2019 Academic Editor: Sergey Dashkovskiy Copyright © 2019 Anding Dai et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper investigates the stability of a class of swarm model with nonlinear dynamics and aperiodically intermittent communication. Different from previous works, it assumes that the agents obtain information from the neighbors at a series of aperiodically time intervals. Moreover, nonlinear dynamics and time delay are considered. It finds that all agents in a swarm can reach cohesion within a finite time under discontinuous communication, where the upper bounds of cohesion depend on the parameters of the swarm model and communication time. A numerical example is given to demonstrate the validity of the theoretical results. 1. Introduction In recent years, distributed cooperative control of multiagents has attracted great interest due to its applications in the fields of physics, engineering, social science, etc. Researchers have spent a lot of efforts in understanding how a herd of animals, school of fish, and man-made mobile autonomous agents can produce coordinated collective behaviors due to local interactions among individuals [1–3]. Swarming is an important branch of distributed coordi- nated control system. Gazi et al. proposed a swarm model which used a special interaction function and studied its aggregation, cohesion, and stability properties [4]. Aſter- wards, the authors [5] proposed an M-member “individual- based” continuous-time swarm model and investigated col- lective behaviors of swarm with general nonlinear attraction and repulsion functions. In [6], stability properties of a continuous-time model for swarm aggregation in the n- dimensional space were discussed, and an asymptotic bound for the spatial size of the swarm was computed by using the parameters of the swarm model. For more about the results of the swarm aggregation problem, see [7, 8] and subsequent references therein. But so far, only a few works have studied swarming behaviors of multiagents with nonlinear dynamical and ape- riodically intermittent communication. Swarm systems with nonlinear dynamics are very common in nature, such as fish school and bird flock, whose trajectories are nonlinear, which means the swarm center of an agent is not a fixed constant. Considering this factor, a class of continuous nonlinear swarm models has been proposed and swarm behaviors have been studied [9]. In [10], stability of nonlinear multiagent systems under directed topology has been researched, which proves that if the topology of the underlying swarm is strongly connected, then all agents will converge globally or exponentially to the super ellipsoid in finite time. In [11, 12], mean square exponential synchronization of dynamic net- works with random switching and parameter uncertainties has been considered, and the conditions for synchronization under strong connected topology are obtained. Consensus of fractional order nonlinear multiagent systems has been studied in [13, 14]. ere is a common feature for achieving stable algorithms of these multiagent systems in [10–14]; that is, each agent can get information from itself and its neighbors without interruption. However, in real life, information about the state of some agents may be unavailable within a certain timetable period; for this reason, intermittent control algorithm was proposed. Each agent can only get information from himself and his neighbors intermittently instead of continuously. At the same time, the intermittent control algorithm has attracted wide attention for it is more effective in saving energy [15–17]. In [14], swarm problem of the Hindawi Mathematical Problems in Engineering Volume 2019, Article ID 3641517, 9 pages https://doi.org/10.1155/2019/3641517
Transcript of Swarming Behaviors in Multiagent Systems with Nonlinear...
Research ArticleSwarming Behaviors in Multiagent Systems with NonlinearDynamics and Aperiodically Intermittent Communication
Anding Dai 12 Yicheng Liu 1 Dongyun Yi1 and Peiying Xiong2
1College of Science National University of Defense Technology Changsha 410073 China2School of Mathematics and Computer Science Hunan City University Yiyang 413000 China
Correspondence should be addressed to Yicheng Liu liuyc2001hotmailcom
Received 9 September 2018 Revised 1 December 2018 Accepted 6 January 2019 Published 23 January 2019
Academic Editor Sergey Dashkovskiy
Copyright copy 2019 Anding Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper investigates the stability of a class of swarm model with nonlinear dynamics and aperiodically intermittentcommunication Different from previous works it assumes that the agents obtain information from the neighbors at a series ofaperiodically time intervals Moreover nonlinear dynamics and time delay are considered It finds that all agents in a swarmcan reach cohesion within a finite time under discontinuous communication where the upper bounds of cohesion depend onthe parameters of the swarm model and communication time A numerical example is given to demonstrate the validity of thetheoretical results
1 Introduction
In recent years distributed cooperative control ofmultiagentshas attracted great interest due to its applications in the fieldsof physics engineering social science etc Researchers havespent a lot of efforts in understanding how a herd of animalsschool of fish and man-made mobile autonomous agentscan produce coordinated collective behaviors due to localinteractions among individuals [1ndash3]
Swarming is an important branch of distributed coordi-nated control system Gazi et al proposed a swarm modelwhich used a special interaction function and studied itsaggregation cohesion and stability properties [4] After-wards the authors [5] proposed an M-member ldquoindividual-basedrdquo continuous-time swarm model and investigated col-lective behaviors of swarm with general nonlinear attractionand repulsion functions In [6] stability properties of acontinuous-time model for swarm aggregation in the n-dimensional space were discussed and an asymptotic boundfor the spatial size of the swarm was computed by using theparameters of the swarm model For more about the resultsof the swarm aggregation problem see [7 8] and subsequentreferences therein
But so far only a few works have studied swarmingbehaviors of multiagents with nonlinear dynamical and ape-riodically intermittent communication Swarm systems with
nonlinear dynamics are very common in nature such as fishschool and bird flock whose trajectories are nonlinear whichmeans the swarm center of an agent is not a fixed constantConsidering this factor a class of continuous nonlinearswarm models has been proposed and swarm behaviors havebeen studied [9] In [10] stability of nonlinear multiagentsystems under directed topology has been researched whichproves that if the topology of the underlying swarm isstrongly connected then all agents will converge globally orexponentially to the super ellipsoid in finite time In [11 12]mean square exponential synchronization of dynamic net-works with random switching and parameter uncertaintieshas been considered and the conditions for synchronizationunder strong connected topology are obtained Consensusof fractional order nonlinear multiagent systems has beenstudied in [13 14] There is a common feature for achievingstable algorithms of these multiagent systems in [10ndash14] thatis each agent can get information from itself and its neighborswithout interruption However in real life informationabout the state of some agents may be unavailable within acertain timetable period for this reason intermittent controlalgorithm was proposed Each agent can only get informationfrom himself and his neighbors intermittently instead ofcontinuously At the same time the intermittent controlalgorithm has attracted wide attention for it is more effectivein saving energy [15ndash17] In [14] swarm problem of the
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3641517 9 pageshttpsdoiorg10115520193641517
2 Mathematical Problems in Engineering
first-order nonlinear multiagent systems under periodicallyintermittent control is discussed and the proportion ofcommunication time to achieve stability is given In [18]consensus of the second-order multiagent systems with peri-odic intermittent communication and nonlinear dynamics isstudied And a new periodic intermittent strategy is proposedto ensure the realization of consensus In [19] consensusproblems for second-order multiagent systems with delayedand intermittent communication were investigated The dis-tributed consensus tracking problem of multiagent systemswith switching directed topologies is studied [20 21] inwhich the control inputs to the followers may be temporallymissed Although many intermittent control strategies havebeen put forward to realize the consensus swarming and syn-chronization of multiagents systems most of the results arebased on the periodic intermittent control And the require-ment of periodicity of intermittent control is quite restrictedIn fact the requirement of periodicity is unreasonable andunnecessary and aperiodically intermittent communicationis more general in many real scenarios [22]
To sum up this paper studies the swarm behaviors ofnonlinear multiagents systems with aperiodically intermit-tent communication to assume the communication betweenagents is intermittent and the communication time is arbi-trary By using Lyapunov approach some sufficient con-ditions are proposed to guarantee the swarm behaviors ofmultiagent systems The contributions of this paper are asfollows
(1) An aperiodically intermittent communication strategyis proposed which ismore general than periodic intermittentcommunication
(2)The nonlinear dynamics and time delay are taken intoaccount
(3) Our conclusion shows that under the strongly con-nected topology the swarm behaviors can be guaranteedwhen the communication ratio reaches a certain condition
2 Model and Preliminaries
The swarm model considered in [4] is first reviewed In aswarm of119873 agents in the 119899-dimensional Euclidean space themotion dynamics of the agent 119894th is described as follows
119894 (119905) = 119873sum119895=1119895 =119894
119892 (119909119894 (119905) minus 119909119895 (119905)) 119894 = 1 2 119873 (1)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905) 119909119894119899(119905)]119879 isin 119877119899 is the positionof agent 119894 at time 119905 119892 119877119899 997888rarr 119877119899 represents theinteraction force between the corresponding agents in theform of repulsion and attraction given by
119892 (119910) = minus119910 [119892119886 (10038171003817100381710038171199101003817100381710038171003817) minus 119892119903 (10038171003817100381710038171199101003817100381710038171003817)] (2)
where 119892119886 119877+ 997888rarr 119877+ represents the attraction term whereas119892119903 119877+ 997888rarr 119877+ represents the repulsion term and 119910 =radic119910119879119910Note that most of the existing protocols are implemented
based on a common assumption that all information istransmitted continuously among agents This assumptionmight be rare case in practice and noneconomic since theinformation exchange among agents does not need to existall the time For example agents may only communicate withtheir neighbors over some disconnected time intervals dueto the unreliability of communication channels failure ofphysical devices etc On the other hand in real biologicalsystems each agentrsquos motion dynamic is determined not onlyby interagent interactions but also by each agentrsquos intrinsicdynamics Therefore the following generalized swarmmodelwith nonlinear dynamics and intermittent communication isconsidered
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(3)
where 119896 = 0 1 2 119891(119905 119909119894(119905) 119909119894(119905 minus 120591)) =[1198911(sdot sdot sdot) 1198912(sdot sdot sdot) 119891119899(sdot sdot sdot)]119879 is continuous vector valuefunctions describing the dynamics of agent 119894 and 120591 is timedelay 119860 = (119886119894119895)119873times119873 is coupled matrix of model (3) if agent119894 is subjected to the action from agent 119895(119894 = 119895) then 119886119894119895 gt 0otherwise 119886119894119895 = 0 Here we take 119886119894119894 = sum119873119895=1119895 =119894 119886119894119895 and assumethat the coupling matrix 119860 is irreducible which means thatthe corresponding digraph is strongly connected [119905119896 119905119896+1)denotes the 119896-th communication period and 119904119896 minus 119905119896 ge 0 isthe communication width
Remark 1 This paper assumes that the communicationtopology of multiagent system (3) is fixed symmetrical and
irreducible In fact the communication topology of thesystem can be further weakened to be switched topology andthe topology contains the case of a directed spanning treeConsistency of multiagent systems with switching topologyand directed spanning trees is studied in references [20 21]However there are few researches on swarming behaviorproblem about this kind of multiagent systems
Remark 2 Thepurpose of this paper is to minimize the com-munication time of agents under the condition of guarantee-ing the swarming behavior of multiagent systems At presentthere are several communication mechanisms in reduc-ing communication time such as impulsive communica-tion [23] intermittent communication and event-triggered
Mathematical Problems in Engineering 3
communication [24] As for event-triggered communicationmechanism the system uses sampled data to determinewhether the triggering condition is satisfied or not If thetriggering condition is satisfied (the plant state deviates morethan a certain threshold from a desired value) the systemcommunicates otherwise it will be no need to communicateCompared with periodic control event-triggered controlhas obvious advantages in resolving communication energycomputing and communication constraints in designingwireless networked control systems [24] In this paper wemainly study the swarming behavior of multiagent systemsunder intermittent communication Unlike event-triggeredcommunication intermittent communication is the inter-mittent exchange of information among agents accordingto the setting communication strategy (ie the interleavingof communication and noncommunication) On the otherhand intermittent communication is related to impulsivecommunicationWhen each communication time119879119896 = 119904119896minus119905119896
tends to zero model (3) evolves into interconnected impul-sive system [23] For the interconnected impulsive system itcan be stabilized by choosing the appropriate pulse intensityand pulse time series (the relevant theories can be consultedin literature [23])
Assume that system (3) satisfies the initial conditions1199090119894 (119905) = [11990901198941(119905) 11990901198942(119905) 1199090119894119899(119905)]119879 isin 119862([1199050 minus 120591 1199050] 119877119899) Todiscuss swarming behaviors we define the set
119909 = 119873sum119894=1
120585119894119909119894 (119905) (4)
as the average position for system (3) where 120585 =(1205851 1205852 120585119873)119879 is the left eigenvector of 119860 corresponding toeigenvalue zero and sum119873119894=1 120585119894 = 1
The dynamical equation of 119909 satisfies
=
119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1) (5)
Define the error vectors 119890119894(119905) = 119909119894(119905) minus 119909 then the errordynamical system can be rewritten as
119890119894 (119905) = 119891(119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869 minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119869 119905 isin [119904119896 119905119896+1)
(6)
where119891(119905 119890119894(119905) 119890119894(119905minus120591)) = 119891(119905 119909119894(119905) 119909119894(119905minus120591))minus119891(119905 119909(119905) 119909(119905minus120591)) and 119869 = 119891(119905 119909(119905) 119909(119905 minus 120591)) minus sum119873119904=1 120585119904119891(119905 119909119904(119905) 119909119904(119905 minus 120591))For the purpose of the swarming behaviors of multiagent
systems (3) we need the following assumptions and lemmas
Assumption 3 The functions 119891(sdot sdot sdot) satisfy the Lipschitzcondition That is there exist two constants 1198971 1198972 gt 0 suchthat1003817100381710038171003817119891 (119905 119909 119909120591) minus 119891 (119905 119910 119910120591)1003817100381710038171003817 le 1198971 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 1198972 1003817100381710038171003817119909120591 minus 1199101205911003817100381710038171003817
forall119909 119910 119909120591 119910120591 isin 119877119899 (7)
Assumption 4 For the aperiodically intermittent communi-cation strategy there exist two positive scalars 0 lt 120579 lt 120596such that for 119894 = 0 1 2
inf119894(119904119894 minus 119905119894) = 120579 gt 0
sup119894(119905119894+1 minus 119905119894) = 120596 lt +infin (8)
Assumption 5 There exist constants 119886 119887 gt 0 such that119892119886(119910) = 119886 119892119903(119910) le 119887119910 for any 119910 isin 119877119899 That is theattraction function is fixed linear and the repulsion functionis bounded
Lemma 6 (see [11]) Let 119909 isin 119877119899 and 119910 isin 119877119899 then for anypositive definite symmetric matrix 119875 isin 119877119899times119899 such thatplusmn2119909119879119910 le 119909119879119875119909 + 119910119879119875minus1119910 (9)
Lemma 7 (see [3]) Given a matrix 119882 = [120596119894119895] sub 119877119872times119872suppose that120596119894119895 ge 0 for 119894 = 119895 and120596119894119894 = minussum119872119895=1119895 =119894 120596119894119895 lt 0 for 119894 =1 2 119872 If119882 is irreducible then119882+119882119879 and 119864119882+119882119879119864are both irreducible where 119864 = diag(1205851 1205852 120585119872) and 120585 =(1205851 1205852 120585119872)119879 are the left eigenvector of119882 corresponding toeigenvalue zero
Lemma 8 (see [25]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that (119905) le minus119886119908 (119905) + 119887max119908119905 (10)
4 Mathematical Problems in Engineering
is satisfied for 119905 ge 120583 If 119886 gt 119887 gt 0 then119908 (119905) le [max119908120583] 119890minus120576(119905minus120583) 119905 ge 120583 (11)
where max119908119905 = sup119905minus120591le119904le119905119908(119904) and 120576 gt 0 is the smallest realroot of the equation
120576 minus 119886 + 119887119890120576120591 = 0 (12)
Lemma 9 (see [18]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that
(119905) le 119886119908 (119905) + 119887max119908119905 (13)
is satisfied for 119905 ge 120583 If 119886 gt 0 119887 gt 0 then119908 (119905) le max119908119905 le [max119908120583] 119890(119886+119887)(119905minus120583) 119905 ge 120583 (14)
wheremax119908119905 = sup119905minus120591le119904le119905119908(119904)3 Main Result
In this section we propose some criteria of swarmingbehaviors for multiagent systems with nonlinear dynamicsand aperiodically intermittent communication
Theorem 10 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus 119905119894 and 120591 le119905119894+1 minus 119904119894 If the following conditions are satisfied(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879
is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1(2) for any 119896 isin 119885+sum119896119894=0minus120582(119904119894minus119905119894)+120574(119905119894+1minus119904119894) le minus120578(119905119896+1minus1199050) where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagent systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (15)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (16)
Proof Choose a nonnegative Lyapunov function as follows
119881 (119905) = 12119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) ) (17)
Then the derivative of 119881(119905) with respect to time 119905 alongthe solutions of error system (6) can be calculated as follows
When 119905 isin [119905119896 119904119896) 119896 = 0 1 2 we get (119905) = 119873sum
119894=1
120585119894119890119879119894 (119905)119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))+ 119873sum119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) = 119873sum119894=1
120585119894119890119879119894 (119905)sdot 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119894=1
120585119894119890119879119894 (119905) 119869minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)]+ 119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])
(18)
Note that sum119873119894=1 120585119894119890119879119894 (119905) = 0 we have the following119873sum119894=1
120585119894119890119879119894 (119905) 119869 = 0 (19)
and
119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])= 0
(20)
By Assumption 3 and Lemma 6 one has
119873sum119894=1
120585119894119890119879119894 (119905) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) le 119873sum119894=1
120585119894 1205762119890119879119894 (119905) 119890119894 (119905)+ 12120576119891119879 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))le ( 1205762 + 119897212120576)
119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591)sdot 119890119894 (119905 minus 120591)
(21)
Mathematical Problems in Engineering 5
According to Assumption 4 and sum119873119895=1 119886119894119895 = 0 we canobtain
minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot 119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)= minus119886 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895 (119890119879119894 (119905) 119890119894 (119905) minus 119890119879119894 (119905) 119890119895 (119905))= minus119886 119873sum119894=1
120585119894( 119873sum119895=1119895 =119894
119886119894119895)119890119879119894 (119905) 119890119894 (119905)
minus 119873sum119895=1119895 =119894
119886119894119895119890119879119894 (119905) 119890119895 (119905) = 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905)
(22)
119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905)) 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)le minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (23)
Substituting (19)-(23) into (18) yields
(119905) le ( 1205762 + 119897212120576)119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)+ 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (24)
Exchanging rows and columns it is not difficult to verifythat
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) = 12119899sum119895=1
119890119879119895 (119905) (119864119860 + 119860119879119864) 119890119895 (119905) (25)
where 119890119895(119905) = [1198901119895(119905) 1198902119895(119905) 119890119873119895(119905)]119879 119864 = diag(12058511205852 120585119873)Consider that the matrix 119860 is irreducible then 119864119860 +119860119879119864 is symmetric irreducible (see Lemma 7) and the sum
of rows is equal to zero There exists an unitary matrix119875 = (1199011 1199012 119901119899) such that 119875119879(119864119860 + 119860119879119864)119875 = Λ whereΛ = diag(1205821 1205822 120582119873) and 0 = 1205821 gt 1205822 gesdot sdot sdot ge 120582119873 Let 119910119895 = 119875119879119890119895(119905) = (1199101198951 1199101198952 119910119895119873)119879 Since1199011 = (1radic119873)[1 1 1]119879 one has 1199101198951 = (1199011)119879119890119895(119905) =(1radic119873)sum119899119894=1 119890119894119895(119905) = 0 Then
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) le 12058222119899sum119895=1
119910119879119895 (119905) 119910119895 (119905)= 12058222
119899sum119895=1
119890119879119895 (119905) 119890119895 (119905)= 12058222
119873sum119894=1
119890119879119894 (119905) 119890119894 (119905) (26)
Let 120585 = max120585119894 | 119894 = 1 2 119873 119889119860 = maxminus119886119894119894 | 119894 =1 2 119873 119897 = 1205762 + 119897212120576 and 120573 = 119897222120576 Note that119873sum119894=1
120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 119873sum119894=1
radic120585119894 1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)1003817100381710038171003817100381710038171003817le ( 119873sum119894=1
120585119894)12 ( 119873sum119894=1
1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)10038171003817100381710038171003817100381710038172)12
= radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(27)
Then taking (26) and (27) into (24) we have
(119905) le (119897 + 11988612058222120585 ) 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119887119889119860radic 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)= minus120572 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + (l + 11988612058222120585 + 120572)sdot radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)
+ 119887119889119860119897 + 11988612058222120585 + 120572) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)
(28)
If sum119873119894=1 120585119894119890119879119894 (119905)119890119894(119905) ge (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 then onehas (119905) le minus2120572119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881 (119904) (29)
Based on Lemma 8 and condition (1) of Theorem 10 thereexists 120582 gt 0 such that 120582 minus 2120572 + 2120573119890120582120591 = 0 and
119881 (119905) le sup119905119896minus120591le119904le119905119896
119881 (119904) 119890minus120582(119905minus119905119896) (30)
Similarly when 119905 isin [119904119896 119905119896+1) we have (119905) le 119897 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)le 2119897119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881(119904) (31)
6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
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2 Mathematical Problems in Engineering
first-order nonlinear multiagent systems under periodicallyintermittent control is discussed and the proportion ofcommunication time to achieve stability is given In [18]consensus of the second-order multiagent systems with peri-odic intermittent communication and nonlinear dynamics isstudied And a new periodic intermittent strategy is proposedto ensure the realization of consensus In [19] consensusproblems for second-order multiagent systems with delayedand intermittent communication were investigated The dis-tributed consensus tracking problem of multiagent systemswith switching directed topologies is studied [20 21] inwhich the control inputs to the followers may be temporallymissed Although many intermittent control strategies havebeen put forward to realize the consensus swarming and syn-chronization of multiagents systems most of the results arebased on the periodic intermittent control And the require-ment of periodicity of intermittent control is quite restrictedIn fact the requirement of periodicity is unreasonable andunnecessary and aperiodically intermittent communicationis more general in many real scenarios [22]
To sum up this paper studies the swarm behaviors ofnonlinear multiagents systems with aperiodically intermit-tent communication to assume the communication betweenagents is intermittent and the communication time is arbi-trary By using Lyapunov approach some sufficient con-ditions are proposed to guarantee the swarm behaviors ofmultiagent systems The contributions of this paper are asfollows
(1) An aperiodically intermittent communication strategyis proposed which ismore general than periodic intermittentcommunication
(2)The nonlinear dynamics and time delay are taken intoaccount
(3) Our conclusion shows that under the strongly con-nected topology the swarm behaviors can be guaranteedwhen the communication ratio reaches a certain condition
2 Model and Preliminaries
The swarm model considered in [4] is first reviewed In aswarm of119873 agents in the 119899-dimensional Euclidean space themotion dynamics of the agent 119894th is described as follows
119894 (119905) = 119873sum119895=1119895 =119894
119892 (119909119894 (119905) minus 119909119895 (119905)) 119894 = 1 2 119873 (1)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905) 119909119894119899(119905)]119879 isin 119877119899 is the positionof agent 119894 at time 119905 119892 119877119899 997888rarr 119877119899 represents theinteraction force between the corresponding agents in theform of repulsion and attraction given by
119892 (119910) = minus119910 [119892119886 (10038171003817100381710038171199101003817100381710038171003817) minus 119892119903 (10038171003817100381710038171199101003817100381710038171003817)] (2)
where 119892119886 119877+ 997888rarr 119877+ represents the attraction term whereas119892119903 119877+ 997888rarr 119877+ represents the repulsion term and 119910 =radic119910119879119910Note that most of the existing protocols are implemented
based on a common assumption that all information istransmitted continuously among agents This assumptionmight be rare case in practice and noneconomic since theinformation exchange among agents does not need to existall the time For example agents may only communicate withtheir neighbors over some disconnected time intervals dueto the unreliability of communication channels failure ofphysical devices etc On the other hand in real biologicalsystems each agentrsquos motion dynamic is determined not onlyby interagent interactions but also by each agentrsquos intrinsicdynamics Therefore the following generalized swarmmodelwith nonlinear dynamics and intermittent communication isconsidered
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(3)
where 119896 = 0 1 2 119891(119905 119909119894(119905) 119909119894(119905 minus 120591)) =[1198911(sdot sdot sdot) 1198912(sdot sdot sdot) 119891119899(sdot sdot sdot)]119879 is continuous vector valuefunctions describing the dynamics of agent 119894 and 120591 is timedelay 119860 = (119886119894119895)119873times119873 is coupled matrix of model (3) if agent119894 is subjected to the action from agent 119895(119894 = 119895) then 119886119894119895 gt 0otherwise 119886119894119895 = 0 Here we take 119886119894119894 = sum119873119895=1119895 =119894 119886119894119895 and assumethat the coupling matrix 119860 is irreducible which means thatthe corresponding digraph is strongly connected [119905119896 119905119896+1)denotes the 119896-th communication period and 119904119896 minus 119905119896 ge 0 isthe communication width
Remark 1 This paper assumes that the communicationtopology of multiagent system (3) is fixed symmetrical and
irreducible In fact the communication topology of thesystem can be further weakened to be switched topology andthe topology contains the case of a directed spanning treeConsistency of multiagent systems with switching topologyand directed spanning trees is studied in references [20 21]However there are few researches on swarming behaviorproblem about this kind of multiagent systems
Remark 2 Thepurpose of this paper is to minimize the com-munication time of agents under the condition of guarantee-ing the swarming behavior of multiagent systems At presentthere are several communication mechanisms in reduc-ing communication time such as impulsive communica-tion [23] intermittent communication and event-triggered
Mathematical Problems in Engineering 3
communication [24] As for event-triggered communicationmechanism the system uses sampled data to determinewhether the triggering condition is satisfied or not If thetriggering condition is satisfied (the plant state deviates morethan a certain threshold from a desired value) the systemcommunicates otherwise it will be no need to communicateCompared with periodic control event-triggered controlhas obvious advantages in resolving communication energycomputing and communication constraints in designingwireless networked control systems [24] In this paper wemainly study the swarming behavior of multiagent systemsunder intermittent communication Unlike event-triggeredcommunication intermittent communication is the inter-mittent exchange of information among agents accordingto the setting communication strategy (ie the interleavingof communication and noncommunication) On the otherhand intermittent communication is related to impulsivecommunicationWhen each communication time119879119896 = 119904119896minus119905119896
tends to zero model (3) evolves into interconnected impul-sive system [23] For the interconnected impulsive system itcan be stabilized by choosing the appropriate pulse intensityand pulse time series (the relevant theories can be consultedin literature [23])
Assume that system (3) satisfies the initial conditions1199090119894 (119905) = [11990901198941(119905) 11990901198942(119905) 1199090119894119899(119905)]119879 isin 119862([1199050 minus 120591 1199050] 119877119899) Todiscuss swarming behaviors we define the set
119909 = 119873sum119894=1
120585119894119909119894 (119905) (4)
as the average position for system (3) where 120585 =(1205851 1205852 120585119873)119879 is the left eigenvector of 119860 corresponding toeigenvalue zero and sum119873119894=1 120585119894 = 1
The dynamical equation of 119909 satisfies
=
119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1) (5)
Define the error vectors 119890119894(119905) = 119909119894(119905) minus 119909 then the errordynamical system can be rewritten as
119890119894 (119905) = 119891(119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869 minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119869 119905 isin [119904119896 119905119896+1)
(6)
where119891(119905 119890119894(119905) 119890119894(119905minus120591)) = 119891(119905 119909119894(119905) 119909119894(119905minus120591))minus119891(119905 119909(119905) 119909(119905minus120591)) and 119869 = 119891(119905 119909(119905) 119909(119905 minus 120591)) minus sum119873119904=1 120585119904119891(119905 119909119904(119905) 119909119904(119905 minus 120591))For the purpose of the swarming behaviors of multiagent
systems (3) we need the following assumptions and lemmas
Assumption 3 The functions 119891(sdot sdot sdot) satisfy the Lipschitzcondition That is there exist two constants 1198971 1198972 gt 0 suchthat1003817100381710038171003817119891 (119905 119909 119909120591) minus 119891 (119905 119910 119910120591)1003817100381710038171003817 le 1198971 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 1198972 1003817100381710038171003817119909120591 minus 1199101205911003817100381710038171003817
forall119909 119910 119909120591 119910120591 isin 119877119899 (7)
Assumption 4 For the aperiodically intermittent communi-cation strategy there exist two positive scalars 0 lt 120579 lt 120596such that for 119894 = 0 1 2
inf119894(119904119894 minus 119905119894) = 120579 gt 0
sup119894(119905119894+1 minus 119905119894) = 120596 lt +infin (8)
Assumption 5 There exist constants 119886 119887 gt 0 such that119892119886(119910) = 119886 119892119903(119910) le 119887119910 for any 119910 isin 119877119899 That is theattraction function is fixed linear and the repulsion functionis bounded
Lemma 6 (see [11]) Let 119909 isin 119877119899 and 119910 isin 119877119899 then for anypositive definite symmetric matrix 119875 isin 119877119899times119899 such thatplusmn2119909119879119910 le 119909119879119875119909 + 119910119879119875minus1119910 (9)
Lemma 7 (see [3]) Given a matrix 119882 = [120596119894119895] sub 119877119872times119872suppose that120596119894119895 ge 0 for 119894 = 119895 and120596119894119894 = minussum119872119895=1119895 =119894 120596119894119895 lt 0 for 119894 =1 2 119872 If119882 is irreducible then119882+119882119879 and 119864119882+119882119879119864are both irreducible where 119864 = diag(1205851 1205852 120585119872) and 120585 =(1205851 1205852 120585119872)119879 are the left eigenvector of119882 corresponding toeigenvalue zero
Lemma 8 (see [25]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that (119905) le minus119886119908 (119905) + 119887max119908119905 (10)
4 Mathematical Problems in Engineering
is satisfied for 119905 ge 120583 If 119886 gt 119887 gt 0 then119908 (119905) le [max119908120583] 119890minus120576(119905minus120583) 119905 ge 120583 (11)
where max119908119905 = sup119905minus120591le119904le119905119908(119904) and 120576 gt 0 is the smallest realroot of the equation
120576 minus 119886 + 119887119890120576120591 = 0 (12)
Lemma 9 (see [18]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that
(119905) le 119886119908 (119905) + 119887max119908119905 (13)
is satisfied for 119905 ge 120583 If 119886 gt 0 119887 gt 0 then119908 (119905) le max119908119905 le [max119908120583] 119890(119886+119887)(119905minus120583) 119905 ge 120583 (14)
wheremax119908119905 = sup119905minus120591le119904le119905119908(119904)3 Main Result
In this section we propose some criteria of swarmingbehaviors for multiagent systems with nonlinear dynamicsand aperiodically intermittent communication
Theorem 10 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus 119905119894 and 120591 le119905119894+1 minus 119904119894 If the following conditions are satisfied(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879
is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1(2) for any 119896 isin 119885+sum119896119894=0minus120582(119904119894minus119905119894)+120574(119905119894+1minus119904119894) le minus120578(119905119896+1minus1199050) where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagent systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (15)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (16)
Proof Choose a nonnegative Lyapunov function as follows
119881 (119905) = 12119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) ) (17)
Then the derivative of 119881(119905) with respect to time 119905 alongthe solutions of error system (6) can be calculated as follows
When 119905 isin [119905119896 119904119896) 119896 = 0 1 2 we get (119905) = 119873sum
119894=1
120585119894119890119879119894 (119905)119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))+ 119873sum119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) = 119873sum119894=1
120585119894119890119879119894 (119905)sdot 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119894=1
120585119894119890119879119894 (119905) 119869minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)]+ 119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])
(18)
Note that sum119873119894=1 120585119894119890119879119894 (119905) = 0 we have the following119873sum119894=1
120585119894119890119879119894 (119905) 119869 = 0 (19)
and
119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])= 0
(20)
By Assumption 3 and Lemma 6 one has
119873sum119894=1
120585119894119890119879119894 (119905) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) le 119873sum119894=1
120585119894 1205762119890119879119894 (119905) 119890119894 (119905)+ 12120576119891119879 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))le ( 1205762 + 119897212120576)
119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591)sdot 119890119894 (119905 minus 120591)
(21)
Mathematical Problems in Engineering 5
According to Assumption 4 and sum119873119895=1 119886119894119895 = 0 we canobtain
minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot 119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)= minus119886 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895 (119890119879119894 (119905) 119890119894 (119905) minus 119890119879119894 (119905) 119890119895 (119905))= minus119886 119873sum119894=1
120585119894( 119873sum119895=1119895 =119894
119886119894119895)119890119879119894 (119905) 119890119894 (119905)
minus 119873sum119895=1119895 =119894
119886119894119895119890119879119894 (119905) 119890119895 (119905) = 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905)
(22)
119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905)) 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)le minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (23)
Substituting (19)-(23) into (18) yields
(119905) le ( 1205762 + 119897212120576)119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)+ 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (24)
Exchanging rows and columns it is not difficult to verifythat
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) = 12119899sum119895=1
119890119879119895 (119905) (119864119860 + 119860119879119864) 119890119895 (119905) (25)
where 119890119895(119905) = [1198901119895(119905) 1198902119895(119905) 119890119873119895(119905)]119879 119864 = diag(12058511205852 120585119873)Consider that the matrix 119860 is irreducible then 119864119860 +119860119879119864 is symmetric irreducible (see Lemma 7) and the sum
of rows is equal to zero There exists an unitary matrix119875 = (1199011 1199012 119901119899) such that 119875119879(119864119860 + 119860119879119864)119875 = Λ whereΛ = diag(1205821 1205822 120582119873) and 0 = 1205821 gt 1205822 gesdot sdot sdot ge 120582119873 Let 119910119895 = 119875119879119890119895(119905) = (1199101198951 1199101198952 119910119895119873)119879 Since1199011 = (1radic119873)[1 1 1]119879 one has 1199101198951 = (1199011)119879119890119895(119905) =(1radic119873)sum119899119894=1 119890119894119895(119905) = 0 Then
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) le 12058222119899sum119895=1
119910119879119895 (119905) 119910119895 (119905)= 12058222
119899sum119895=1
119890119879119895 (119905) 119890119895 (119905)= 12058222
119873sum119894=1
119890119879119894 (119905) 119890119894 (119905) (26)
Let 120585 = max120585119894 | 119894 = 1 2 119873 119889119860 = maxminus119886119894119894 | 119894 =1 2 119873 119897 = 1205762 + 119897212120576 and 120573 = 119897222120576 Note that119873sum119894=1
120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 119873sum119894=1
radic120585119894 1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)1003817100381710038171003817100381710038171003817le ( 119873sum119894=1
120585119894)12 ( 119873sum119894=1
1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)10038171003817100381710038171003817100381710038172)12
= radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(27)
Then taking (26) and (27) into (24) we have
(119905) le (119897 + 11988612058222120585 ) 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119887119889119860radic 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)= minus120572 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + (l + 11988612058222120585 + 120572)sdot radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)
+ 119887119889119860119897 + 11988612058222120585 + 120572) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)
(28)
If sum119873119894=1 120585119894119890119879119894 (119905)119890119894(119905) ge (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 then onehas (119905) le minus2120572119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881 (119904) (29)
Based on Lemma 8 and condition (1) of Theorem 10 thereexists 120582 gt 0 such that 120582 minus 2120572 + 2120573119890120582120591 = 0 and
119881 (119905) le sup119905119896minus120591le119904le119905119896
119881 (119904) 119890minus120582(119905minus119905119896) (30)
Similarly when 119905 isin [119904119896 119905119896+1) we have (119905) le 119897 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)le 2119897119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881(119904) (31)
6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
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Mathematical Problems in Engineering 3
communication [24] As for event-triggered communicationmechanism the system uses sampled data to determinewhether the triggering condition is satisfied or not If thetriggering condition is satisfied (the plant state deviates morethan a certain threshold from a desired value) the systemcommunicates otherwise it will be no need to communicateCompared with periodic control event-triggered controlhas obvious advantages in resolving communication energycomputing and communication constraints in designingwireless networked control systems [24] In this paper wemainly study the swarming behavior of multiagent systemsunder intermittent communication Unlike event-triggeredcommunication intermittent communication is the inter-mittent exchange of information among agents accordingto the setting communication strategy (ie the interleavingof communication and noncommunication) On the otherhand intermittent communication is related to impulsivecommunicationWhen each communication time119879119896 = 119904119896minus119905119896
tends to zero model (3) evolves into interconnected impul-sive system [23] For the interconnected impulsive system itcan be stabilized by choosing the appropriate pulse intensityand pulse time series (the relevant theories can be consultedin literature [23])
Assume that system (3) satisfies the initial conditions1199090119894 (119905) = [11990901198941(119905) 11990901198942(119905) 1199090119894119899(119905)]119879 isin 119862([1199050 minus 120591 1199050] 119877119899) Todiscuss swarming behaviors we define the set
119909 = 119873sum119894=1
120585119894119909119894 (119905) (4)
as the average position for system (3) where 120585 =(1205851 1205852 120585119873)119879 is the left eigenvector of 119860 corresponding toeigenvalue zero and sum119873119894=1 120585119894 = 1
The dynamical equation of 119909 satisfies
=
119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119873sum119894=1
120585119894119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1) (5)
Define the error vectors 119890119894(119905) = 119909119894(119905) minus 119909 then the errordynamical system can be rewritten as
119890119894 (119905) = 119891(119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869 minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119869 119905 isin [119904119896 119905119896+1)
(6)
where119891(119905 119890119894(119905) 119890119894(119905minus120591)) = 119891(119905 119909119894(119905) 119909119894(119905minus120591))minus119891(119905 119909(119905) 119909(119905minus120591)) and 119869 = 119891(119905 119909(119905) 119909(119905 minus 120591)) minus sum119873119904=1 120585119904119891(119905 119909119904(119905) 119909119904(119905 minus 120591))For the purpose of the swarming behaviors of multiagent
systems (3) we need the following assumptions and lemmas
Assumption 3 The functions 119891(sdot sdot sdot) satisfy the Lipschitzcondition That is there exist two constants 1198971 1198972 gt 0 suchthat1003817100381710038171003817119891 (119905 119909 119909120591) minus 119891 (119905 119910 119910120591)1003817100381710038171003817 le 1198971 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 1198972 1003817100381710038171003817119909120591 minus 1199101205911003817100381710038171003817
forall119909 119910 119909120591 119910120591 isin 119877119899 (7)
Assumption 4 For the aperiodically intermittent communi-cation strategy there exist two positive scalars 0 lt 120579 lt 120596such that for 119894 = 0 1 2
inf119894(119904119894 minus 119905119894) = 120579 gt 0
sup119894(119905119894+1 minus 119905119894) = 120596 lt +infin (8)
Assumption 5 There exist constants 119886 119887 gt 0 such that119892119886(119910) = 119886 119892119903(119910) le 119887119910 for any 119910 isin 119877119899 That is theattraction function is fixed linear and the repulsion functionis bounded
Lemma 6 (see [11]) Let 119909 isin 119877119899 and 119910 isin 119877119899 then for anypositive definite symmetric matrix 119875 isin 119877119899times119899 such thatplusmn2119909119879119910 le 119909119879119875119909 + 119910119879119875minus1119910 (9)
Lemma 7 (see [3]) Given a matrix 119882 = [120596119894119895] sub 119877119872times119872suppose that120596119894119895 ge 0 for 119894 = 119895 and120596119894119894 = minussum119872119895=1119895 =119894 120596119894119895 lt 0 for 119894 =1 2 119872 If119882 is irreducible then119882+119882119879 and 119864119882+119882119879119864are both irreducible where 119864 = diag(1205851 1205852 120585119872) and 120585 =(1205851 1205852 120585119872)119879 are the left eigenvector of119882 corresponding toeigenvalue zero
Lemma 8 (see [25]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that (119905) le minus119886119908 (119905) + 119887max119908119905 (10)
4 Mathematical Problems in Engineering
is satisfied for 119905 ge 120583 If 119886 gt 119887 gt 0 then119908 (119905) le [max119908120583] 119890minus120576(119905minus120583) 119905 ge 120583 (11)
where max119908119905 = sup119905minus120591le119904le119905119908(119904) and 120576 gt 0 is the smallest realroot of the equation
120576 minus 119886 + 119887119890120576120591 = 0 (12)
Lemma 9 (see [18]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that
(119905) le 119886119908 (119905) + 119887max119908119905 (13)
is satisfied for 119905 ge 120583 If 119886 gt 0 119887 gt 0 then119908 (119905) le max119908119905 le [max119908120583] 119890(119886+119887)(119905minus120583) 119905 ge 120583 (14)
wheremax119908119905 = sup119905minus120591le119904le119905119908(119904)3 Main Result
In this section we propose some criteria of swarmingbehaviors for multiagent systems with nonlinear dynamicsand aperiodically intermittent communication
Theorem 10 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus 119905119894 and 120591 le119905119894+1 minus 119904119894 If the following conditions are satisfied(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879
is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1(2) for any 119896 isin 119885+sum119896119894=0minus120582(119904119894minus119905119894)+120574(119905119894+1minus119904119894) le minus120578(119905119896+1minus1199050) where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagent systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (15)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (16)
Proof Choose a nonnegative Lyapunov function as follows
119881 (119905) = 12119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) ) (17)
Then the derivative of 119881(119905) with respect to time 119905 alongthe solutions of error system (6) can be calculated as follows
When 119905 isin [119905119896 119904119896) 119896 = 0 1 2 we get (119905) = 119873sum
119894=1
120585119894119890119879119894 (119905)119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))+ 119873sum119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) = 119873sum119894=1
120585119894119890119879119894 (119905)sdot 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119894=1
120585119894119890119879119894 (119905) 119869minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)]+ 119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])
(18)
Note that sum119873119894=1 120585119894119890119879119894 (119905) = 0 we have the following119873sum119894=1
120585119894119890119879119894 (119905) 119869 = 0 (19)
and
119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])= 0
(20)
By Assumption 3 and Lemma 6 one has
119873sum119894=1
120585119894119890119879119894 (119905) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) le 119873sum119894=1
120585119894 1205762119890119879119894 (119905) 119890119894 (119905)+ 12120576119891119879 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))le ( 1205762 + 119897212120576)
119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591)sdot 119890119894 (119905 minus 120591)
(21)
Mathematical Problems in Engineering 5
According to Assumption 4 and sum119873119895=1 119886119894119895 = 0 we canobtain
minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot 119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)= minus119886 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895 (119890119879119894 (119905) 119890119894 (119905) minus 119890119879119894 (119905) 119890119895 (119905))= minus119886 119873sum119894=1
120585119894( 119873sum119895=1119895 =119894
119886119894119895)119890119879119894 (119905) 119890119894 (119905)
minus 119873sum119895=1119895 =119894
119886119894119895119890119879119894 (119905) 119890119895 (119905) = 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905)
(22)
119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905)) 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)le minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (23)
Substituting (19)-(23) into (18) yields
(119905) le ( 1205762 + 119897212120576)119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)+ 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (24)
Exchanging rows and columns it is not difficult to verifythat
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) = 12119899sum119895=1
119890119879119895 (119905) (119864119860 + 119860119879119864) 119890119895 (119905) (25)
where 119890119895(119905) = [1198901119895(119905) 1198902119895(119905) 119890119873119895(119905)]119879 119864 = diag(12058511205852 120585119873)Consider that the matrix 119860 is irreducible then 119864119860 +119860119879119864 is symmetric irreducible (see Lemma 7) and the sum
of rows is equal to zero There exists an unitary matrix119875 = (1199011 1199012 119901119899) such that 119875119879(119864119860 + 119860119879119864)119875 = Λ whereΛ = diag(1205821 1205822 120582119873) and 0 = 1205821 gt 1205822 gesdot sdot sdot ge 120582119873 Let 119910119895 = 119875119879119890119895(119905) = (1199101198951 1199101198952 119910119895119873)119879 Since1199011 = (1radic119873)[1 1 1]119879 one has 1199101198951 = (1199011)119879119890119895(119905) =(1radic119873)sum119899119894=1 119890119894119895(119905) = 0 Then
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) le 12058222119899sum119895=1
119910119879119895 (119905) 119910119895 (119905)= 12058222
119899sum119895=1
119890119879119895 (119905) 119890119895 (119905)= 12058222
119873sum119894=1
119890119879119894 (119905) 119890119894 (119905) (26)
Let 120585 = max120585119894 | 119894 = 1 2 119873 119889119860 = maxminus119886119894119894 | 119894 =1 2 119873 119897 = 1205762 + 119897212120576 and 120573 = 119897222120576 Note that119873sum119894=1
120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 119873sum119894=1
radic120585119894 1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)1003817100381710038171003817100381710038171003817le ( 119873sum119894=1
120585119894)12 ( 119873sum119894=1
1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)10038171003817100381710038171003817100381710038172)12
= radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(27)
Then taking (26) and (27) into (24) we have
(119905) le (119897 + 11988612058222120585 ) 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119887119889119860radic 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)= minus120572 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + (l + 11988612058222120585 + 120572)sdot radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)
+ 119887119889119860119897 + 11988612058222120585 + 120572) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)
(28)
If sum119873119894=1 120585119894119890119879119894 (119905)119890119894(119905) ge (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 then onehas (119905) le minus2120572119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881 (119904) (29)
Based on Lemma 8 and condition (1) of Theorem 10 thereexists 120582 gt 0 such that 120582 minus 2120572 + 2120573119890120582120591 = 0 and
119881 (119905) le sup119905119896minus120591le119904le119905119896
119881 (119904) 119890minus120582(119905minus119905119896) (30)
Similarly when 119905 isin [119904119896 119905119896+1) we have (119905) le 119897 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)le 2119897119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881(119904) (31)
6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
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4 Mathematical Problems in Engineering
is satisfied for 119905 ge 120583 If 119886 gt 119887 gt 0 then119908 (119905) le [max119908120583] 119890minus120576(119905minus120583) 119905 ge 120583 (11)
where max119908119905 = sup119905minus120591le119904le119905119908(119904) and 120576 gt 0 is the smallest realroot of the equation
120576 minus 119886 + 119887119890120576120591 = 0 (12)
Lemma 9 (see [18]) Let 119908 [120583 minus 120591 +infin) 997888rarr [0 +infin) be acontinuous function such that
(119905) le 119886119908 (119905) + 119887max119908119905 (13)
is satisfied for 119905 ge 120583 If 119886 gt 0 119887 gt 0 then119908 (119905) le max119908119905 le [max119908120583] 119890(119886+119887)(119905minus120583) 119905 ge 120583 (14)
wheremax119908119905 = sup119905minus120591le119904le119905119908(119904)3 Main Result
In this section we propose some criteria of swarmingbehaviors for multiagent systems with nonlinear dynamicsand aperiodically intermittent communication
Theorem 10 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus 119905119894 and 120591 le119905119894+1 minus 119904119894 If the following conditions are satisfied(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879
is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1(2) for any 119896 isin 119885+sum119896119894=0minus120582(119904119894minus119905119894)+120574(119905119894+1minus119904119894) le minus120578(119905119896+1minus1199050) where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagent systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (15)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (16)
Proof Choose a nonnegative Lyapunov function as follows
119881 (119905) = 12119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) ) (17)
Then the derivative of 119881(119905) with respect to time 119905 alongthe solutions of error system (6) can be calculated as follows
When 119905 isin [119905119896 119904119896) 119896 = 0 1 2 we get (119905) = 119873sum
119894=1
120585119894119890119879119894 (119905)119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))+ 119873sum119895=1119895 =119894
119886119894119895119892 (119890119894 (119905) minus 119890119895 (119905)) + 119869minus 119873sum119904=1
119873sum119895=1
120585119904119886119904119895119892 (119890119904 (119905) minus 119890119895 (119905)) = 119873sum119894=1
120585119894119890119879119894 (119905)sdot 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) + 119873sum
119894=1
120585119894119890119879119894 (119905) 119869minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)]+ 119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])
(18)
Note that sum119873119894=1 120585119894119890119879119894 (119905) = 0 we have the following119873sum119894=1
120585119894119890119879119894 (119905) 119869 = 0 (19)
and
119873sum119894=1
120585119894119890119879119894 (119905) ( 119873sum119904=1
119873sum119895=1
120585119904119886119904119895 (119890119904 (119905) minus 119890119895 (119905))sdot [119892119886 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817) minus 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)])= 0
(20)
By Assumption 3 and Lemma 6 one has
119873sum119894=1
120585119894119890119879119894 (119905) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) le 119873sum119894=1
120585119894 1205762119890119879119894 (119905) 119890119894 (119905)+ 12120576119891119879 (119905 119890119894 (119905) 119890119894 (119905 minus 120591)) 119891 (119905 119890119894 (119905) 119890119894 (119905 minus 120591))le ( 1205762 + 119897212120576)
119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591)sdot 119890119894 (119905 minus 120591)
(21)
Mathematical Problems in Engineering 5
According to Assumption 4 and sum119873119895=1 119886119894119895 = 0 we canobtain
minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot 119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)= minus119886 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895 (119890119879119894 (119905) 119890119894 (119905) minus 119890119879119894 (119905) 119890119895 (119905))= minus119886 119873sum119894=1
120585119894( 119873sum119895=1119895 =119894
119886119894119895)119890119879119894 (119905) 119890119894 (119905)
minus 119873sum119895=1119895 =119894
119886119894119895119890119879119894 (119905) 119890119895 (119905) = 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905)
(22)
119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905)) 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)le minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (23)
Substituting (19)-(23) into (18) yields
(119905) le ( 1205762 + 119897212120576)119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)+ 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (24)
Exchanging rows and columns it is not difficult to verifythat
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) = 12119899sum119895=1
119890119879119895 (119905) (119864119860 + 119860119879119864) 119890119895 (119905) (25)
where 119890119895(119905) = [1198901119895(119905) 1198902119895(119905) 119890119873119895(119905)]119879 119864 = diag(12058511205852 120585119873)Consider that the matrix 119860 is irreducible then 119864119860 +119860119879119864 is symmetric irreducible (see Lemma 7) and the sum
of rows is equal to zero There exists an unitary matrix119875 = (1199011 1199012 119901119899) such that 119875119879(119864119860 + 119860119879119864)119875 = Λ whereΛ = diag(1205821 1205822 120582119873) and 0 = 1205821 gt 1205822 gesdot sdot sdot ge 120582119873 Let 119910119895 = 119875119879119890119895(119905) = (1199101198951 1199101198952 119910119895119873)119879 Since1199011 = (1radic119873)[1 1 1]119879 one has 1199101198951 = (1199011)119879119890119895(119905) =(1radic119873)sum119899119894=1 119890119894119895(119905) = 0 Then
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) le 12058222119899sum119895=1
119910119879119895 (119905) 119910119895 (119905)= 12058222
119899sum119895=1
119890119879119895 (119905) 119890119895 (119905)= 12058222
119873sum119894=1
119890119879119894 (119905) 119890119894 (119905) (26)
Let 120585 = max120585119894 | 119894 = 1 2 119873 119889119860 = maxminus119886119894119894 | 119894 =1 2 119873 119897 = 1205762 + 119897212120576 and 120573 = 119897222120576 Note that119873sum119894=1
120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 119873sum119894=1
radic120585119894 1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)1003817100381710038171003817100381710038171003817le ( 119873sum119894=1
120585119894)12 ( 119873sum119894=1
1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)10038171003817100381710038171003817100381710038172)12
= radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(27)
Then taking (26) and (27) into (24) we have
(119905) le (119897 + 11988612058222120585 ) 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119887119889119860radic 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)= minus120572 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + (l + 11988612058222120585 + 120572)sdot radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)
+ 119887119889119860119897 + 11988612058222120585 + 120572) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)
(28)
If sum119873119894=1 120585119894119890119879119894 (119905)119890119894(119905) ge (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 then onehas (119905) le minus2120572119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881 (119904) (29)
Based on Lemma 8 and condition (1) of Theorem 10 thereexists 120582 gt 0 such that 120582 minus 2120572 + 2120573119890120582120591 = 0 and
119881 (119905) le sup119905119896minus120591le119904le119905119896
119881 (119904) 119890minus120582(119905minus119905119896) (30)
Similarly when 119905 isin [119904119896 119905119896+1) we have (119905) le 119897 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)le 2119897119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881(119904) (31)
6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
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Mathematical Problems in Engineering 5
According to Assumption 4 and sum119873119895=1 119886119894119895 = 0 we canobtain
minus 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905))sdot 119892119886 (10038171003817100381710038171003817119890119894 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)= minus119886 119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895 (119890119879119894 (119905) 119890119894 (119905) minus 119890119879119894 (119905) 119890119895 (119905))= minus119886 119873sum119894=1
120585119894( 119873sum119895=1119895 =119894
119886119894119895)119890119879119894 (119905) 119890119894 (119905)
minus 119873sum119895=1119895 =119894
119886119894119895119890119879119894 (119905) 119890119895 (119905) = 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905)
(22)
119873sum119894=1
119873sum119895=1119895 =119894
120585119894119886119894119895119890119879119894 (119905) (119890119894 (119905) minus 119890119895 (119905)) 119892119903 (10038171003817100381710038171003817119890119904 (119905) minus 119890119895 (119905)10038171003817100381710038171003817)le minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (23)
Substituting (19)-(23) into (18) yields
(119905) le ( 1205762 + 119897212120576)119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119897222120576119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)+ 119886 119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) minus 119873sum119894=1
119887119886119894119894120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 (24)
Exchanging rows and columns it is not difficult to verifythat
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) = 12119899sum119895=1
119890119879119895 (119905) (119864119860 + 119860119879119864) 119890119895 (119905) (25)
where 119890119895(119905) = [1198901119895(119905) 1198902119895(119905) 119890119873119895(119905)]119879 119864 = diag(12058511205852 120585119873)Consider that the matrix 119860 is irreducible then 119864119860 +119860119879119864 is symmetric irreducible (see Lemma 7) and the sum
of rows is equal to zero There exists an unitary matrix119875 = (1199011 1199012 119901119899) such that 119875119879(119864119860 + 119860119879119864)119875 = Λ whereΛ = diag(1205821 1205822 120582119873) and 0 = 1205821 gt 1205822 gesdot sdot sdot ge 120582119873 Let 119910119895 = 119875119879119890119895(119905) = (1199101198951 1199101198952 119910119895119873)119879 Since1199011 = (1radic119873)[1 1 1]119879 one has 1199101198951 = (1199011)119879119890119895(119905) =(1radic119873)sum119899119894=1 119890119894119895(119905) = 0 Then
119873sum119894=1
119873sum119895=1
120585119894119886119894119895119890119879119894 (119905) 119890119895 (119905) le 12058222119899sum119895=1
119910119879119895 (119905) 119910119895 (119905)= 12058222
119899sum119895=1
119890119879119895 (119905) 119890119895 (119905)= 12058222
119873sum119894=1
119890119879119894 (119905) 119890119894 (119905) (26)
Let 120585 = max120585119894 | 119894 = 1 2 119873 119889119860 = maxminus119886119894119894 | 119894 =1 2 119873 119897 = 1205762 + 119897212120576 and 120573 = 119897222120576 Note that119873sum119894=1
120585119894 1003817100381710038171003817119890119894 (119905)1003817100381710038171003817 = 119873sum119894=1
radic120585119894 1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)1003817100381710038171003817100381710038171003817le ( 119873sum119894=1
120585119894)12 ( 119873sum119894=1
1003817100381710038171003817100381710038171003817radic120585119894119890119894 (119905)10038171003817100381710038171003817100381710038172)12
= radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(27)
Then taking (26) and (27) into (24) we have
(119905) le (119897 + 11988612058222120585 ) 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)+ 119887119889119860radic 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)= minus120572 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + (l + 11988612058222120585 + 120572)sdot radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)(radic 119873sum119894=1
120585119894119890119879119894 (119905) 119890119894 (119905)
+ 119887119889119860119897 + 11988612058222120585 + 120572) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)
(28)
If sum119873119894=1 120585119894119890119879119894 (119905)119890119894(119905) ge (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 then onehas (119905) le minus2120572119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881 (119904) (29)
Based on Lemma 8 and condition (1) of Theorem 10 thereexists 120582 gt 0 such that 120582 minus 2120572 + 2120573119890120582120591 = 0 and
119881 (119905) le sup119905119896minus120591le119904le119905119896
119881 (119904) 119890minus120582(119905minus119905119896) (30)
Similarly when 119905 isin [119904119896 119905119896+1) we have (119905) le 119897 119873sum
119894=1
120585119894119890119879119894 (119905) 119890119894 (119905) + 120573 119873sum119894=1
120585119894119890119879119894 (119905 minus 120591) 119890119894 (119905 minus 120591)le 2119897119881 (119905) + 2120573 sup
119905minus120591le119904le119905119881(119904) (31)
6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
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6 Mathematical Problems in Engineering
Define119882(119905) = 119890120582119905119881(119905) based on (30) and (31) one has
(119905) le 120582119882(119905) + 2119897119882 (119905) + 2120573119890120582120591 sup119905minus120591le119904le119905
119882(119904) 119905 isin [119904119896 119905119896+1) (32)
According to Lemma 9 and (30) we have
119882(119905) le sup119904119896minus120591le119904le119904119896
119882(119904) 119890(120582+120574)(119905minus119904119896)= sup119904119896minus120591le119904le119904119896
119881 (119904) 119890120582119904119890(120582+120574)(119905minus119904119896)le sup119904119896minus120591le119904le119904119896
sup119905119896minus120591le120579le119905119896
119881 (120579) 119890minus120582(119904minus119905119896)119890120582119904119890(120582+120574)(119905minus119904119896)= sup119905119896minus120591le120579le119905119896
119881(120579) 119890120582119905119896+(120582+120574)(119905minus119904119896)
(33)
where 120574 = 2(119897 + 120573119890120582120591)Hence
119881(119905) le sup119905119896minus120591le119904le119905119896
119881(119904) 119890minus120582(119904119896minus119905119896)+120574(119905minus119904119896) 119905 isin [119904119896 119905119896+1) (34)
By induction we have
119881 (119905)le sup119905119896minus120591le119904le119905119896
sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1)+120574(119904minus119904119896minus1)sdot 119890minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896)le sup119905119896minus1minus120591le120579le119905119896minus1
119881 (120579) 119890minus120582(119904119896minus1minus119905119896minus1+119904119896minus119905119896)+120574(119905119896minus119904119896minus1+119905119896+1minus119904119896)le sdot sdot sdot le sup
1199050minus120591le120579le1199050
119881(120579) 119890minus120582sum119896119894=1(119904119894minus119905119894)+120574sum119896119894=1(119905119894+1minus119904119894)
(35)
According to condition (2) in Theorem 10 we can get
119881 (119905) le sup1199050minus120591le120579le1199050
119881 (120579) 119890minus120578(119905minus1199050) forall119905 isin [119905119896 119905119896+1) (36)
This implies that the swarm will globally and exponen-tially converge to the hyperellipsoid 119861120598 Furthermore allagentswillmove into the hyperball119861120598 in a finite time specifiedby
119905 le 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (37)
The proof is complete
Remark 11 In Theorem 10 we have given a sufficient con-dition to guarantee multiagent systems to achieve swarmingbehaviors under intermittent communication It needs tobe pointed out that under the premise that the guaranteecondition 2 is established communication or noncommuni-cation is allowed in some communication period It means
that minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) = 120574(119905119894+1 minus 119905119894) or minus120582(119904119894 minus119905119894) + 120574(119905119894+1 minus 119904119894) = minus120582(119905119894+1 minus 119905119894) holds in some period Inaddition it can be seen from the proof of Theorem 10 thatin the time period [119905119894 119904119894) there is communication betweenagents which is beneficial to system stability In the timeperiod [119904119894 119905119894+1) there is no communication between agentswhich is not beneficial (possibly harmful) to system stabilityTherefore in order to ensure the swarm the communicationtime between agents should be as long as possible but thenoncommunication time should be as short as possibleOn the other hand condition (2) in Theorem 10 is easyto achieve when minus120582(119904119894 minus 119905119894) + 120574(119905119894+1 minus 119904119894) le minus120578(119905119896+1 minus119905119894) holds in all periods For any 119894 = 0 1 2 if 119904119894 minus119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 then aperiodically intermittentcommunication becomes the case of periodically intermittentcommunication
In the following we will verify (35) for some special cases
Corollary 12 Let Assumptions 3ndash5 hold 120591 le 119904119894 minus119905119894 and 120591 le 119905119894+1 minus 119904119894 If the following conditions aresatisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582(119904119896minus119905119896)+120574(119905119896+1minus119904119896) le minus120578(119905119896+1minus119905119896)where 120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (38)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (39)
For any 119894 = 0 1 2 if 119904119894 minus 119905119894 equiv 119879119888 119905119894+1 minus 119904119894 equiv 119879119906119888 where119879119888 and 119879119906119888 are both positive constants We have the followingresult
Corollary 13 Let Assumptions 3ndash5 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576 120573 =119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879 isthe left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 7
then all the agents of multiagent systems (3) will convergeto a hyperball 119861120598 centered at 119909(119905)
119861120598 = (1199091 119909119873) | 119873sum119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (40)
where 120598 = (119887119889119860)2(119897 + 11988612058222120585 + 120572)2 Furthermore all agentswill move into the hyperball 119861120598 in a finite time specified by
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881 (120579)) (41)
Next we use the special class of attraction and repulsionfunction to analyze the swarming behaviors of multiagentssystems with aperiodically intermittent communication Ittakes the form
119892 (119910) = minus119910(119886 minus 119887 exp(minus1003817100381710038171003817y10038171003817100381710038172c
) (42)
where 119887 gt 119886 gt 0 and 119888 gt 0 are constants We can derive thefollowing result
Theorem 14 Let Assumptions 3-4 hold 120591 le 119879119888 and 120591 le 119879119906119888If the following conditions are satisfied
(1) there exist positive constants 120572 120582 such that 119897 +11988612058222120585+120572 le 0 and 120582 minus 2120572 + 2120573119890120582120591 = 0 hold where 119897 = 1205762 + 119897212120576120573 = 119897222120576 120585 = max120585119894 | 119894 = 1 2 119873 120585 = (1205851 1205852 120585119873)119879is the left eigenvector corresponding to eigenvalues 0 of 119860 andsum119873119894=1 120585119894 = 1
(2) for any 119896 isin 119885+ minus120582119879119888 + 120574119879119906119888 le minus120578(119879119906 + 119879119906119888) where120574 = 2(119897 + 120573119890120582120591) and 120578 gt 0then all the agents of multiagents systems (3) will converge
to a hyperball 119861120598 centered at 119909(119905)119861120598 = (1199091 119909119873) | 119873sum
119894=1
120585119894 1003817100381710038171003817119909119894 (119905) minus 119909 (119905)10038171003817100381710038172 le 120598 (43)
where 120598 = (119887119889119860radic119888)22119890(119897 + 11988612058222120585 + 120572)2 Furthermore allagents will move into the hyperball 119861120598 in a finite time specifiedby
119905 = 1199050 minus 1120578 ln( 120598sup1199050minus120591le120579le1199050119881(120579)) (44)
Proof It is easy to see that 119892119886(119910) = 119886Define 119865(119909) = 119887119909119890minus1199092119888 then
1198651015840 (119909) = 119887(1 minus 21199092119888 ) 119890minus1199092119888 = gt 0 119909 lt radic 1198882 lt 0 119909 gt radic 1198882
(45)
Hence
119865 (119909) le 119865(radic 1198882) = 119887radic1198882 119890minus12 (46)
So
10038171003817100381710038171199101003817100381710038171003817 119892119903 (10038171003817100381710038171199101003817100381710038171003817) le 119887radic1198882 119890minus12 (47)
Therefore Assumption 5 holds when 119892(119910) is the same as(42)
It is known from Corollary 13 that Theorem 14 is estab-lished
The proof is complete
4 Numerical Simulation
In this section we will use a numerical simulation to illustratethe feasibility and effectiveness of our results
Consider the swarmmodel consisting of six agents whichis described as follows
119894 (119905) = 119891(119905 119909119894 (119905) 119909119894 (119905 minus 120591)) + 119873sum
119895=1119895 =119894
119886119894119895119892 (119909119894 (119905) minus 119909119895 (119905)) 119905 isin [119905119896 119904119896) 119891 (119905 119909119894 (119905) 119909119894 (119905 minus 120591)) 119905 isin [119904119896 119905119896+1)
(48)
where 119909119894(119905) = [1199091198941(119905) 1199091198942(119905)]119879 119891(sdot sdot sdot) = tanh(119909119894(119905)) +02 tanh(119909119894(119905 minus 120591)) 119892(119909119894 minus 119909119895) = minus(119909119894 minus 119909119895)(119886 minus 119887119890minus119909119894minus1199091198952119888)and 120591 = 1 119886 = 1 119887 = 20 119888 = 02 and the coupled matric ischosen as
119860 =[[[[[[[[[[[[
minus5 1 1 1 1 11 minus5 1 1 1 11 1 minus5 1 1 11 1 1 minus5 1 11 1 1 1 minus5 11 1 1 1 1 minus5
]]]]]]]]]]]]
(49)
The nonlinear function 119891(sdot sdot sdot) satisfies the Lipschitzcondition with 1198971 = 1 1198972 = 02 It is easy to know that 1205822 = minus5and 120585 = 16 according to the coupled matrix 119860 Choose120576 = 1 and 120572 = 4 then we get 120582 = 44779 120574 = 55221Let the ratio of communication time to noncommunicationtime be greater than 13376 then all conditions ofTheorem 14are satisfied and 120578 gt 02 Let the initial states of theagents be chosen randomly from [minus10 10] Moreover thecommunication period 119879 = 3119904 and for the communicationwidth see Figure 1 (119910 = 1)The results of simulation are givenin Figure 2 It is clear that under the aperiodically intermittentcommunication all agents can achieve swarming behavior
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
0
02
04
06
08
1
Figure 1 Communication (119910 = 1) and noncommunication (119910 = 0)time of the system (48)
minus6 minus4 minus2 0 2 4 6 8 10minus20
minus15
minus10
minus5
0
5
10Agentsrsquo trajectories on the planＲ1minusＲ
2
Ｒ1
Ｒ2
Figure 2 The trajectories of agents in 1199091-1199092 plan
5 Conclusions
In this paper the swarm stability of nonlinear multiagentsystems with aperiodically intermittent communication isstudied Each agent gets information from its neighbors ina series of aperiodically intervals In addition each agenthas nonlinear dynamics and time delay It is shown in thispaper that under aperiodic discontinuous communication allagents in a swarm can reach cohesion within a finite timeand the upper bounds of cohesion depend on the parametersof the swarm model the second largest eigenvalue of thecoupling matrix and communication time Furthermore anexample is given to verify the correctness of the theoreticalresults
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofChina (11671011) the Natural Science Foundation of Hunan
Province of China (2016JJ4016 2016JJ6019) and the Educa-tional Department of Hunan Province of China (17B049)
References
[1] MH Degroot ldquoReaching a consensusrdquo Journal of the AmericanStatistical Association vol 69 no 345 pp 118ndash121 1974
[2] J Buhl D J T Sumpter I D Couzin et al ldquoFrom disorder toorder in marching locustsrdquo Science vol 312 no 5778 pp 1402ndash1406 2006
[3] R Olfati-Saber and R M Murray ldquoConsensus problems innetworks of agents with switching topology and time-delaysrdquoIEEETransactions on Automatic Control vol 49 no 9 pp 1520ndash1533 2004
[4] V Gazi and K M Passino ldquoStability analysis of swarmsrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 48 no 4 pp 692ndash697 2003
[5] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004
[6] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004
[7] N E Leonard and E Fiorelli ldquoVirtual leaders artificial poten-tials and coordinated control of groupsrdquo in Proceedings of theConference Decision Control pp 2968ndash2973 2001
[8] Y Liu and K M Passino ldquoStable social foraging swarms ina noisy environmentrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 1 pp30ndash44 2004
[9] W Yu G Chen M Cao J Lu and H-T Zhang ldquoSwarmingbehaviors in multi-agent systems with nonlinear dynamicsrdquoChaos vol 23 no 4 Article ID 043118 2013
[10] W Li ldquoStability analysis of swarms with general topologyrdquo IEEETransactions on SystemsMan andCybernetics vol 38 no 4 pp1084ndash1097 2008
[11] Y Xu W Zhou J Fang C Xie and D Tong ldquoFinite-timesynchronization of the complex dynamical network with non-derivative and derivative couplingrdquo Neurocomputing vol 173pp 1356ndash1361 2016
[12] A DaiW Zhou Y Xu and C Xiao ldquoAdaptive exponential syn-chronization in mean square for Markovian jumping neutral-type coupled neural networks with time-varying delays bypinning controlrdquo Neurocomputing vol 173 pp 809ndash818 2016
[13] P Gong and W Lan ldquoAdaptive robust tracking control foruncertain nonlinear fractional-order multi-agent systems withdirected topologiesrdquo Automatica vol 92 pp 92ndash99 2018
[14] FWang and Y Yang ldquoLeader-following consensus of nonlinearfractional-order multi-agent systems via event-triggered con-trolrdquo International Journal of Systems Science vol 48 no 3 pp571ndash577 2017
[15] X Liu and T Chen ldquoCluster synchronization in directednetworks via intermittent pinning controlrdquo IEEE Transactionson Neural Networks and Learning Systems vol 22 no 7 pp1009ndash1020 2011
[16] Y Lei Y Wang H Chen and X Liu ldquoDistributed controlof heterogeneous linear multi-agent systems by intermittentevent-triggered controlrdquo Automation pp 34ndash39 2017
[17] H Li Y Zhu J Wang et al ldquoConsensus of nonlinear second-order multi-agent systems with mixed time-delays and inter-mittent communicationsrdquo Neurocomputing vol 251 pp 115ndash126 2017
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
[18] GWen ZDuanWYu andGChen ldquoSecond-order consensusof multi-agent systems with delayed nonlinear dynamics andintermittent measurementsrdquo International Journal of Controlvol 86 no 2 pp 322ndash331 2013
[19] N Huang Z Duan and Y Zhao ldquoConsensus of multi-agentsystems via delayed and intermittent communicationsrdquo IETControlTheory andApplications vol 8 no 10 pp 782ndash795 2015
[20] G Wen Z Duan G Chen and W Yu ldquoConsensus trackingof multi-agent systems with Lipschitz-type node dynamicsand switching topologiesrdquo IEEE Transactions on Circuits andSystems I Regular Papers vol 61 no 2 pp 499ndash511 2014
[21] G Wen G Hu W Yu and J Cao ldquoConsensus trackingfor higher-order multi-agent systems with switching directedtopologies and occasionally missing control inputsrdquo Systemsand Control Letters vol 62 no 12 pp 1151ndash1158 2013
[22] Z Yu H Jiang C Hu and X Fan ldquoConsensus of second-order multi-agent systems with delayed nonlinear dynamicsand aperiodically intermittent communicationsrdquo InternationalJournal of Control vol 90 no 5 pp 909ndash922 2017
[23] S Dashkovskiy and P Feketa ldquoInput-to-state stability of impul-sive systems and their networksrdquo Nonlinear Analysis HybridSystems vol 26 pp 190ndash200 2017
[24] W Heemels K Johansson and P Tabuada ldquoAn introduction toevent-triggered and self-triggered controlrdquo in Proceedings of theIEEE Conference on Decision and Control pp 3270ndash3285 2012
[25] A Halanay Differential Equations Stability Oscillations TimeLags Academic Press California Calif USA 1966
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
