Research Article The Exponential Stability Result of an...
11
Research Article The Exponential Stability Result of an Euler-Bernoulli Beam Equation with Interior Delays and Boundary Damping Peng-cheng Han, Yan-fang Li, Gen-qi Xu, and Dan-hong Liu Department of Mathematics, Tianjin University, Tianjin 300350, China Correspondence should be addressed to Yan-fang Li; [email protected] Received 20 January 2016; Accepted 9 March 2016 Academic Editor: Honglei Xu Copyright © 2016 Peng-cheng Han 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. We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping. At first, we prove the well-posedness of the system by the 0 semigroup theory. Next we study the exponential stability of the system by constructing appropriate Lyapunov functionals. We transform the exponential stability issue into the solvability of inequality equations. By analyzing the relationship between delays parameters and damping parameters , we describe (, )-region for which the system is exponentially stable. Furthermore, we obtain an estimation of the decay rate ∗ . 1. Introduction It is well known that the time delay always exists in real system, which may be caused by acquisition of response and excitation data, online data processing, and computation of control forces. Since time delay may destroy stability [1, 2] even if it is very small, the stabilization problem of systems with time delays has been a hot topic in the mathematical control theory and engineering. In recent years, the systems described by PDEs with time delays have been an active area of research; see [3–7] and references therein. Generally speaking, there are mainly three kinds of time delay in the system, one is the interior time delay of the system (also called structural memory), one is the input delay (control delay), and the third is the output delay (measurement delay). Many scholars have made great efforts to minimize the negative effects of time delays although time delay cannot be eliminated due to its inherent nature, for example, [8–10] for boundary control with delays, [11, 12] for internal control delays, and [13] for output delays. In past several years, the research on the Euler-Bernoulli beam with time delay has made great progress. For example, Park et al. [14] considered the stabilization problem of an Euler-Bernoulli beam with structural memory; Liang et al. [15] introduced the modified Smith predictor to Euler- Bernoulli beam with the boundary control and the delayed boundary measurement; Shang et al. [16–18] investigated the stabilization problem of the Euler-Bernoulli beam with boundary input delay; Yang et al. [19, 20] solved the stabi- lization problem of constant and variable coefficients Euler- Bernoulli beam with delayed observation and boundary control; at the same time, Jin and Guo [21] solved the output feedback stabilization of Euler-Bernoulli beam by Lyapunov approach. However, few people investigate the influence of an Euler-Bernoulli beam with interior delays and boundary damping on the system stability. In this paper we mainly study the exponential stability of a system described by the Euler- Bernoulli beam with interior delays and boundary damping. More precisely, we consider the following system, whose dynamic behavior is governed by the Euler-Bernoulli beam: (, ) + (, ) − 2 (, − ) = 0, ∈ (0, 1) , > 0, (0, ) = (0, ) = (1, ) = 0, (1, ) = (1, ) , (, 0) = 0 () , (, 0) = 1 () , ∈ (0, 1) , (, ) = ℎ 0 (, ) , ∈ (0, 1) , ∈ (−, 0) , (1) Hindawi Publishing Corporation Journal of Difference Equations Volume 2016, Article ID 3732176, 10 pages http://dx.doi.org/10.1155/2016/3732176
Transcript of Research Article The Exponential Stability Result of an...
Research ArticleThe Exponential Stability Result of an Euler-Bernoulli BeamEquation with Interior Delays and Boundary Damping
Peng-cheng Han Yan-fang Li Gen-qi Xu and Dan-hong Liu
Department of Mathematics Tianjin University Tianjin 300350 China
Correspondence should be addressed to Yan-fang Li fangyan123litjueducn
Received 20 January 2016 Accepted 9 March 2016
Academic Editor Honglei Xu
Copyright copy 2016 Peng-cheng Han et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We study the exponential stability of Euler-Bernoulli beam with interior time delays and boundary damping At first we prove thewell-posedness of the system by the 119862
0semigroup theory Next we study the exponential stability of the system by constructing
appropriate Lyapunov functionals We transform the exponential stability issue into the solvability of inequality equations Byanalyzing the relationship between delays parameters 120572 and damping parameters 120573 we describe (120573 120572)-region for which the systemis exponentially stable Furthermore we obtain an estimation of the decay rate 120582lowast
1 Introduction
It is well known that the time delay always exists in realsystem which may be caused by acquisition of response andexcitation data online data processing and computation ofcontrol forces Since time delay may destroy stability [1 2]even if it is very small the stabilization problem of systemswith time delays has been a hot topic in the mathematicalcontrol theory and engineering In recent years the systemsdescribed by PDEs with time delays have been an activearea of research see [3ndash7] and references therein Generallyspeaking there are mainly three kinds of time delay in thesystem one is the interior time delay of the system (alsocalled structural memory) one is the input delay (controldelay) and the third is the output delay (measurement delay)Many scholars have made great efforts to minimize thenegative effects of time delays although time delay cannotbe eliminated due to its inherent nature for example [8ndash10]for boundary control with delays [11 12] for internal controldelays and [13] for output delays
In past several years the research on the Euler-Bernoullibeam with time delay has made great progress For examplePark et al [14] considered the stabilization problem of anEuler-Bernoulli beam with structural memory Liang etal [15] introduced the modified Smith predictor to Euler-Bernoulli beam with the boundary control and the delayed
boundary measurement Shang et al [16ndash18] investigatedthe stabilization problem of the Euler-Bernoulli beam withboundary input delay Yang et al [19 20] solved the stabi-lization problem of constant and variable coefficients Euler-Bernoulli beam with delayed observation and boundarycontrol at the same time Jin and Guo [21] solved the outputfeedback stabilization of Euler-Bernoulli beam by Lyapunovapproach However few people investigate the influence ofan Euler-Bernoulli beam with interior delays and boundarydamping on the system stability In this paperwemainly studythe exponential stability of a system described by the Euler-Bernoulli beam with interior delays and boundary dampingMore precisely we consider the following system whosedynamic behavior is governed by the Euler-Bernoulli beam
119910119905119905(119909 119905) + 119910
119909119909119909119909(119909 119905) minus 2120572119910
119905(119909 119905 minus 120591) = 0
119909 isin (0 1) 119905 gt 0
119910 (0 119905) = 119910119909(0 119905) = 119910
119909119909(1 119905) = 0
119910119909119909119909
(1 119905) = 120573119910119905(1 119905)
119910 (119909 0) = 1199100(119909)
119910119905(119909 0) = 119910
1(119909)
119909 isin (0 1)
119910119905(119909 119904) = ℎ
0(119909 119904) 119909 isin (0 1) 119904 isin (minus120591 0)
(1)
Hindawi Publishing CorporationJournal of Difference EquationsVolume 2016 Article ID 3732176 10 pageshttpdxdoiorg10115520163732176
2 Journal of Difference Equations
with 120573 120572 120591 gt 0 where 119910119905(119909 119905) = 120597119910120597119905 119910
119909(119909 119905) = 120597119910120597119909
and 120591 is the delay time We mainly investigate its exponentialstability
The rest is organized as follows In Section 2 we at firstformulate problem (1) into an appropriate Hilbert space Hand then study the well-posedness of the system by thesemigroup theory In Section 3 we construct a Lyapunovfunctional for system (1) and prove the exponential stabilityunder certain conditions By optimization parameters weobtain a complicated relationship between the decay rate120582 and delay time 120591 Finally in Section 4 we give a briefconclusion
2 Well-Posedness of the System
In this section we will discuss the well-posedness and somebasic properties of system (1) For the purpose firstly weformulate system (1) into an appropriate Hilbert space
Set
119911 (119909 120588 119905) = 119910119905(119909 119905 minus 120591120588)
119909 isin (0 1) 120588 isin (0 1) 119905 gt 0(2)
Clearly 119911(119909 120588 119905) satisfies
120591119911119905(119909 120588 119905) + 119911
120588(119909 120588 119905) = 0 119909 isin (0 1) 120588 isin (0 1)
119911 (119909 0 119905) = 119910119905 (119909 119905)
119911 (119909 1 119905) = 119910119905(119909 119905 minus 120591)
(3)
Thus system (1) is equivalent to the following
119910119905119905(119909 119905) + 119910
119909119909119909119909(119909 119905) minus 2120572119911 (119909 1 119905) = 0 119909 isin (0 1)
120591119911119905(119909 120588 119905) + 119911
120588(119909 120588 119905) = 0 119909 isin (0 1) 120588 (0 1)
119910 (0 119905) = 119910119909(0 119905) = 119910
119909119909(1 119905) = 0
119910119909119909119909
(1 119905) = 120573119910119905(1 119905)
119911 (119909 0 119905) = 119910119905(119909 119905)
119910 (119909 0) = 1199100(119909)
119910119905 (119909 119905) = 119910
1 (119909)
119911 (119909 120588 0) = 1199110(119909 120588) = ℎ
0(119909 minus120591120588)
(4)
Set
1198672
119864(0 1) = 119910 isin 119867
2(0 1) | 119910 (0) = 119910
1015840(0) = 0 (5)
where119867119896(0 1) is the usual Sobolev space of order 119896 We takethe state space as
H = 1198672
119864(0 1) times 119871
2(0 1) times 119871
2[(0 1) times (0 1)] (6)
equipped with the following inner product for any 119884119894=
(119891119894 119892119894 ℎ119894)119879isin H 119894 = 1 2
⟨1198841 1198842⟩H= int
1
0
(1198911119909119909
(119909) 1198912119909119909
(119909) + 1198921(119909) 1198922(119909)) 119889119909
+ 120591∬
1
0
ℎ1(119909 120588) ℎ
2(119909 120588)119889120588 119889119909
(7)
Obviously (H sdot H) is a Hilbert spaceWe define an operatorA inH by
A(
119891
119892
ℎ
) = (
119892 (119909)
minus119891119909119909119909119909
(119909) + 2120572ℎ (119909 1)
minus1
120591ℎ120588(119909 120588)
) (8)
with domain
119863 (A) = (119891 119892 ℎ)119879isin H
1003816100381610038161003816100381610038161003816100381610038161003816
119891 isin 1198674[0 1] cap 119867
2
119864[0 1] 119892 isin 119867
2
119864(0 1) ℎ isin 119867
1(0 1)
11989110158401015840(1) = 0 119891
101584010158401015840(1) = 120573119892 (1) ℎ (119909 0) = 119892 (119909)
(9)
With the assistance of operator A we can rewrite (4) as anevolution equation inH
119889119884 (119905)
119889119905= A119884 (119905) 119905 gt 0
119884 (0) = 1198840
(10)
where 119884(119905) = (119910(119909 119905) 119910119905(119909 119905) 119911(119909 120588 119905))
119879 and 1198840= (1199100(119909)
1199101(119909) 1199110(119909 120588))
119879For operatorA we have the following result
Lemma 1 LetA be defined as (8) and (9) ThenA is a closedand densely defined linear operator in H For any 120573 gt 0 and
120572 gt 0 0 isin 120588(119860) and Aminus1 is compact on H Hence 120590(A)
consists of all isolated eigenvalues of finite multiplicity
Proof It is easy to check that A is a closed and denselydefined linear operator in H the detail of the verification isomitted
Let 120572 gt 0 120573 gt 0 and for any 119865 ≜ (120583 ] 120596)119879 isin H weconsider the equation A119884 = 119865 where 119884 = (119891 119892 ℎ) isin 119863(A)that is
119892 (119909) = 120583 (119909)
minus119891119909119909119909119909
(119909) + 2120572ℎ (119909 1) = ] (119909)
Journal of Difference Equations 3
minus1
120591ℎ120588(119909 120588) = 120596 (119909 120588)
(11)
with boundary condition
119891 (0) = 1198911015840(0) = 119891
10158401015840(1) = 0
119891101584010158401015840(1) = 120573119892 (1)
ℎ (119909 0) = 119892 (119909)
(12)
By a complex calculation we get the solution
119892 (119909) = 120583 (119909)
ℎ (119909 120588) = 120583 (119909) minus int
120588
0
120591120596 (119909 119904) 119889119904
119891 (119909) = minusint
119909
0
int
119903
0
int
119902
1
int
119901
1
(2120572120583 (119904) minus 2120572int
1
0
120591120596 (119904 119896) 119889119896 minus 120592 (119904)) 119889119904 119889119901 119889119902 119889119903 +1
6120573120583 (1) 119909
3
(13)
Let 119891 119892 ℎ be given as (13) Then we have A119884 = 119865 and119884 = (119891 119892 ℎ) isin 119863(A) The closed operator theorem assertsthat 0 isin 120588(119860) and Aminus1 H rarr 119863(A) is a bounded linearoperator Since 119863(A) sub 119867
4
119864(0 1) times 119867
2(0 1) times 119867
1(0 1) the
Sobolev Embedding Theorem asserts that Aminus1 is a compactoperator on H Hence by the spectral theory of compactoperator 120590(A) consists of all isolated eigenvalues of finitemultiplicity
Theorem 2 Let A and H be defined as before Then Agenerates a 119862
0semigroup on H Hence system (10) is well
posed
Proof For any real 119884 = (119891 119892 ℎ)119879isin D(A) we calculate
⟨A119884 119884⟩H = int
1
0
11989210158401015840(119909) 11989110158401015840(119909) 119889119909
+ int
1
0
119892 (119909) (minus119891119909119909119909119909 (119909) + 2120572ℎ (119909 1)) 119889119909
minus∬
1
0
ℎ120588(119909 120588) ℎ (119909 120588) 119889119909 119889120588
= minus119892 (1) 119891101584010158401015840(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
(ℎ2(119909 1) minus ℎ
2(119909 0)) 119889119909
= minus1205731198922(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
ℎ2(119909 1) 119889119909 +
1
2int
1
0
1198922(119909) 119889119909
(14)
Since 120573 gt 0 we have
⟨A119884 119884⟩H le minus1
2int
1
0
(ℎ (119909 1) minus 2120572119892 (119909))2119889119909
+1
2int
1
0
(41205722+ 1) 119892
2(119909) 119889119909
le 119872⟨119884 119884⟩H
(15)
where119872 = 21205722+12 which shows thatAminus119872119868 is a dissipative
operator This together with Lemma 1 shows that A minus 119872119868
satisfies the conditions of Lumer-Phillips theorem [22] SoAgenerates a 119862
0semigroup onH
3 Exponential Stability of the System
In this section we consider the exponential stability issue ofsystem (1) based on Lyapunov method
The energy function of system (1) is defined as
119864 (119905) =1
2int
1
0
[1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)] 119889119909 (16)
In what follows we will give some lemmas that are thefoundation of our method
Lemma 3 (see [23]) Let 119864(119905) be a nonnegative function onR+ If there exists a function 119881(119905) and some positive numbers
1198881and 120582 such that the conditions
119881 (119905) gt 1198881119890120582119905119864 (119905) forall119905 ge 0 (17)
(119905) le 0 forall119905 ge 0 (18)
hold then 119864(119905) decays exponentially at rate 120582
In order to construct a function 119881(119905) satisfying theconditions in Lemma 3 we set
119866 (119905) = 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909 (19)
where 120578 is a constant and satisfies 0 lt 120578 lt 2
We can establish an equivalence relation between 119866(119905)
and 119864(119905) via the following Lemma
Lemma 4 Let 119864(119905) and 119866(119905) be defined as before Then thereexist positive constants 119888
2and 1198883such that
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) forall119905 ge 0 (20)
holds
4 Journal of Difference Equations
Proof Let 119910(119909 119905) be the solution of (1) Applying Youngrsquos andPoincarersquos inequalities100381610038161003816100381610038161003816100381610038161003816
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
le120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
11990921199102
119909(119909 119905) 119889119909
le120575
2int
1
0
1199091199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
1199092119889119909int
119909
0
1199102
119909119909(119904 119905) 119889119904
lt120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
8120575int
1
0
1199102
119909119909(119909 119905) 119889119909
forall120575 gt 0 119905 ge 0
(21)
Taking 120575 = 12 we get100381610038161003816100381610038161003816100381610038161003816
120578 int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
lt120578
2119864 (119905) 119905 ge 0 (22)
Since 0 lt 120578 lt 2 we can set 1198881= 1 minus 1205782 and 119888
2= 1 + 1205782
then
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) (23)
The desired inequality follows
Let 120582 gt 0 We define a function 119881(119905) by
119881 (119905) = 1198811(119905) + 119881
2(119905) (24)
where
1198811(119905) = 119890
2120582119905(1
2int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909)
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909
(25)
Noting that 120572 gt 0 according to Lemma 4 we can see that thefollowing result is true
Lemma 5 Let 119881(119905) defined as before Then 119881(119905) satisfiescondition (17) that is
119881 (119905) gt 11988821198902120582119905119864 (119905) 119905 ge 0 (26)
In what follows we calculate (119905) For 1198811(119905) we have the
following result
Lemma 6 Let 1198811(119905) be defined as before and let 119910(119909 119905) be the
solution of (1) Then
1 (119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)
sdot int
1
0
1199102
119909119909119889119909 + (120582 minus
120578
2+120582120578
2+ 120572119890120582120591)
sdot int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573 +
120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
119890120582119905int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(27)
Proof By definition we see that
1198811(119905) = 119890
2120582119905(119864 (119905) + 119866 (119905)) (28)
where 119864(119905) and 119866(119905) are defined as beforeSo
1 (119905) = 119890
2120582119905(2120582 (119864 (119905) + 119866 (119905)) + (119905) + (119905)) (29)
In what follows we will calculate (119905) and (119905)Using integration by parts and the boundary condition
we have
(119905) = int
1
0
119910119909119909 (119909 119905) 119910119909119909119905 (119909 119905) + 119910119905 (119909 119905) 119910119905119905 (119909 119905) 119889119909
= 119910119909119909 (119909 119905) 119910119905119909 (119909 119905)
1003816100381610038161003816
1
0
minus int
1
0
119910119909119909119909 (119909 119905) 119910119905119909 (119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ int
1
0
2120572119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
= 119910119909119909(119909 119905) 119910
119905119909(119909 119905)
1003816100381610038161003816
1
0minus 119910119909119909119909
(119909 119905) 119910119905(119909 119905)
1003816100381610038161003816
1
0
+ int
1
0
119910119909119909119909119909
(119909 119905) 119910119905(119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909 = minus120573119910
2
119905(1 119905)
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
(119905) = 120578 (int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ int
1
0
119909119910119909(119909 119905) 119910
119905119905(119909 119905) 119889119909)
= 120578(int
1
0
119909119910119905119909 (119909 119905) 119910119905 (119909 119905) 119889119909
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
Journal of Difference Equations 5
minus int
1
0
119909119910119909 (119909 119905) 119910119909119909119909 (119909 119905) 119889119909) = 120578(
1
21199102
119905(1 119905)
minus1
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3
2int
1
0
1199102
119909119909(119909 119905) 119889119909
minus 120573119910119909(1 119905) 119910
119905(1 119905)
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909)
(30)where we have used equalities
int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
=1
21199102
119905(1 119905) minus
1
2int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909119909119909119909
(119909 119905) 119910119909(119909 119905) 119889119909
=3
2int
1
0
1199102
119909119909(119909 119905) 119889119909 + 120573119910
119909(1 119905) 119910
119905(1 119905)
(31)
Summarizing the above all we have
1= 1198902120582119905
[120582int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909] + 119890
2120582119905[minus1205731199102
119905(1 119905)
+120578
21199102
119905(1 119905) minus 120573120578119910119909 (1 119905) 119910119905 (1 119905)
minus120578
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
= 1198902120582119905
[(120582 minus3120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
+ 1198902120582119905
[120578 minus 2120573
21199102
119905(1 119905) minus 120573120578119910
119909(1 119905) 119910
119905(1 119905)]
(32)
Since
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
le1
4int
1
0
1199102
119909119909119889119909 +
1
4int
1
0
1199102
119905(119909 119905) 119889119909
minus 120573120578119910119909(1 119905) 119910
119905(1 119905) le
120573120578
21199102
119909(1 119905) +
120573120578
21199102
119905(1 119905)
le120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909 +
120573120578
21199102
119905(1 119905)
(33)
we have
1le 1198902120582119905
[(120582 minus3120578
2+120582120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909] + 119890
2120582119905 120578 minus 2120573
2
sdot 1199102
119905(1 119905) + 119890
2120582119905[120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+120573120578
21199102
119905(1 119905)]
= 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909] + 119890
2120582119905
sdot120578 minus 2120573 + 120573120578
21199102
119905(1 119905)
(34)
We now estimate the integral terms with time delay ApplyingYoungrsquos and Poincarersquos inequalities we have
int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205751
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
21205751
int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205752
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
81205752
int
1
0
1199102
119909119909(119909 119905) 119889119909
(35)
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Difference Equations
with 120573 120572 120591 gt 0 where 119910119905(119909 119905) = 120597119910120597119905 119910
119909(119909 119905) = 120597119910120597119909
and 120591 is the delay time We mainly investigate its exponentialstability
The rest is organized as follows In Section 2 we at firstformulate problem (1) into an appropriate Hilbert space Hand then study the well-posedness of the system by thesemigroup theory In Section 3 we construct a Lyapunovfunctional for system (1) and prove the exponential stabilityunder certain conditions By optimization parameters weobtain a complicated relationship between the decay rate120582 and delay time 120591 Finally in Section 4 we give a briefconclusion
2 Well-Posedness of the System
In this section we will discuss the well-posedness and somebasic properties of system (1) For the purpose firstly weformulate system (1) into an appropriate Hilbert space
Set
119911 (119909 120588 119905) = 119910119905(119909 119905 minus 120591120588)
119909 isin (0 1) 120588 isin (0 1) 119905 gt 0(2)
Clearly 119911(119909 120588 119905) satisfies
120591119911119905(119909 120588 119905) + 119911
120588(119909 120588 119905) = 0 119909 isin (0 1) 120588 isin (0 1)
119911 (119909 0 119905) = 119910119905 (119909 119905)
119911 (119909 1 119905) = 119910119905(119909 119905 minus 120591)
(3)
Thus system (1) is equivalent to the following
119910119905119905(119909 119905) + 119910
119909119909119909119909(119909 119905) minus 2120572119911 (119909 1 119905) = 0 119909 isin (0 1)
120591119911119905(119909 120588 119905) + 119911
120588(119909 120588 119905) = 0 119909 isin (0 1) 120588 (0 1)
119910 (0 119905) = 119910119909(0 119905) = 119910
119909119909(1 119905) = 0
119910119909119909119909
(1 119905) = 120573119910119905(1 119905)
119911 (119909 0 119905) = 119910119905(119909 119905)
119910 (119909 0) = 1199100(119909)
119910119905 (119909 119905) = 119910
1 (119909)
119911 (119909 120588 0) = 1199110(119909 120588) = ℎ
0(119909 minus120591120588)
(4)
Set
1198672
119864(0 1) = 119910 isin 119867
2(0 1) | 119910 (0) = 119910
1015840(0) = 0 (5)
where119867119896(0 1) is the usual Sobolev space of order 119896 We takethe state space as
H = 1198672
119864(0 1) times 119871
2(0 1) times 119871
2[(0 1) times (0 1)] (6)
equipped with the following inner product for any 119884119894=
(119891119894 119892119894 ℎ119894)119879isin H 119894 = 1 2
⟨1198841 1198842⟩H= int
1
0
(1198911119909119909
(119909) 1198912119909119909
(119909) + 1198921(119909) 1198922(119909)) 119889119909
+ 120591∬
1
0
ℎ1(119909 120588) ℎ
2(119909 120588)119889120588 119889119909
(7)
Obviously (H sdot H) is a Hilbert spaceWe define an operatorA inH by
A(
119891
119892
ℎ
) = (
119892 (119909)
minus119891119909119909119909119909
(119909) + 2120572ℎ (119909 1)
minus1
120591ℎ120588(119909 120588)
) (8)
with domain
119863 (A) = (119891 119892 ℎ)119879isin H
1003816100381610038161003816100381610038161003816100381610038161003816
119891 isin 1198674[0 1] cap 119867
2
119864[0 1] 119892 isin 119867
2
119864(0 1) ℎ isin 119867
1(0 1)
11989110158401015840(1) = 0 119891
101584010158401015840(1) = 120573119892 (1) ℎ (119909 0) = 119892 (119909)
(9)
With the assistance of operator A we can rewrite (4) as anevolution equation inH
119889119884 (119905)
119889119905= A119884 (119905) 119905 gt 0
119884 (0) = 1198840
(10)
where 119884(119905) = (119910(119909 119905) 119910119905(119909 119905) 119911(119909 120588 119905))
119879 and 1198840= (1199100(119909)
1199101(119909) 1199110(119909 120588))
119879For operatorA we have the following result
Lemma 1 LetA be defined as (8) and (9) ThenA is a closedand densely defined linear operator in H For any 120573 gt 0 and
120572 gt 0 0 isin 120588(119860) and Aminus1 is compact on H Hence 120590(A)
consists of all isolated eigenvalues of finite multiplicity
Proof It is easy to check that A is a closed and denselydefined linear operator in H the detail of the verification isomitted
Let 120572 gt 0 120573 gt 0 and for any 119865 ≜ (120583 ] 120596)119879 isin H weconsider the equation A119884 = 119865 where 119884 = (119891 119892 ℎ) isin 119863(A)that is
119892 (119909) = 120583 (119909)
minus119891119909119909119909119909
(119909) + 2120572ℎ (119909 1) = ] (119909)
Journal of Difference Equations 3
minus1
120591ℎ120588(119909 120588) = 120596 (119909 120588)
(11)
with boundary condition
119891 (0) = 1198911015840(0) = 119891
10158401015840(1) = 0
119891101584010158401015840(1) = 120573119892 (1)
ℎ (119909 0) = 119892 (119909)
(12)
By a complex calculation we get the solution
119892 (119909) = 120583 (119909)
ℎ (119909 120588) = 120583 (119909) minus int
120588
0
120591120596 (119909 119904) 119889119904
119891 (119909) = minusint
119909
0
int
119903
0
int
119902
1
int
119901
1
(2120572120583 (119904) minus 2120572int
1
0
120591120596 (119904 119896) 119889119896 minus 120592 (119904)) 119889119904 119889119901 119889119902 119889119903 +1
6120573120583 (1) 119909
3
(13)
Let 119891 119892 ℎ be given as (13) Then we have A119884 = 119865 and119884 = (119891 119892 ℎ) isin 119863(A) The closed operator theorem assertsthat 0 isin 120588(119860) and Aminus1 H rarr 119863(A) is a bounded linearoperator Since 119863(A) sub 119867
4
119864(0 1) times 119867
2(0 1) times 119867
1(0 1) the
Sobolev Embedding Theorem asserts that Aminus1 is a compactoperator on H Hence by the spectral theory of compactoperator 120590(A) consists of all isolated eigenvalues of finitemultiplicity
Theorem 2 Let A and H be defined as before Then Agenerates a 119862
0semigroup on H Hence system (10) is well
posed
Proof For any real 119884 = (119891 119892 ℎ)119879isin D(A) we calculate
⟨A119884 119884⟩H = int
1
0
11989210158401015840(119909) 11989110158401015840(119909) 119889119909
+ int
1
0
119892 (119909) (minus119891119909119909119909119909 (119909) + 2120572ℎ (119909 1)) 119889119909
minus∬
1
0
ℎ120588(119909 120588) ℎ (119909 120588) 119889119909 119889120588
= minus119892 (1) 119891101584010158401015840(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
(ℎ2(119909 1) minus ℎ
2(119909 0)) 119889119909
= minus1205731198922(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
ℎ2(119909 1) 119889119909 +
1
2int
1
0
1198922(119909) 119889119909
(14)
Since 120573 gt 0 we have
⟨A119884 119884⟩H le minus1
2int
1
0
(ℎ (119909 1) minus 2120572119892 (119909))2119889119909
+1
2int
1
0
(41205722+ 1) 119892
2(119909) 119889119909
le 119872⟨119884 119884⟩H
(15)
where119872 = 21205722+12 which shows thatAminus119872119868 is a dissipative
operator This together with Lemma 1 shows that A minus 119872119868
satisfies the conditions of Lumer-Phillips theorem [22] SoAgenerates a 119862
0semigroup onH
3 Exponential Stability of the System
In this section we consider the exponential stability issue ofsystem (1) based on Lyapunov method
The energy function of system (1) is defined as
119864 (119905) =1
2int
1
0
[1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)] 119889119909 (16)
In what follows we will give some lemmas that are thefoundation of our method
Lemma 3 (see [23]) Let 119864(119905) be a nonnegative function onR+ If there exists a function 119881(119905) and some positive numbers
1198881and 120582 such that the conditions
119881 (119905) gt 1198881119890120582119905119864 (119905) forall119905 ge 0 (17)
(119905) le 0 forall119905 ge 0 (18)
hold then 119864(119905) decays exponentially at rate 120582
In order to construct a function 119881(119905) satisfying theconditions in Lemma 3 we set
119866 (119905) = 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909 (19)
where 120578 is a constant and satisfies 0 lt 120578 lt 2
We can establish an equivalence relation between 119866(119905)
and 119864(119905) via the following Lemma
Lemma 4 Let 119864(119905) and 119866(119905) be defined as before Then thereexist positive constants 119888
2and 1198883such that
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) forall119905 ge 0 (20)
holds
4 Journal of Difference Equations
Proof Let 119910(119909 119905) be the solution of (1) Applying Youngrsquos andPoincarersquos inequalities100381610038161003816100381610038161003816100381610038161003816
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
le120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
11990921199102
119909(119909 119905) 119889119909
le120575
2int
1
0
1199091199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
1199092119889119909int
119909
0
1199102
119909119909(119904 119905) 119889119904
lt120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
8120575int
1
0
1199102
119909119909(119909 119905) 119889119909
forall120575 gt 0 119905 ge 0
(21)
Taking 120575 = 12 we get100381610038161003816100381610038161003816100381610038161003816
120578 int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
lt120578
2119864 (119905) 119905 ge 0 (22)
Since 0 lt 120578 lt 2 we can set 1198881= 1 minus 1205782 and 119888
2= 1 + 1205782
then
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) (23)
The desired inequality follows
Let 120582 gt 0 We define a function 119881(119905) by
119881 (119905) = 1198811(119905) + 119881
2(119905) (24)
where
1198811(119905) = 119890
2120582119905(1
2int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909)
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909
(25)
Noting that 120572 gt 0 according to Lemma 4 we can see that thefollowing result is true
Lemma 5 Let 119881(119905) defined as before Then 119881(119905) satisfiescondition (17) that is
119881 (119905) gt 11988821198902120582119905119864 (119905) 119905 ge 0 (26)
In what follows we calculate (119905) For 1198811(119905) we have the
following result
Lemma 6 Let 1198811(119905) be defined as before and let 119910(119909 119905) be the
solution of (1) Then
1 (119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)
sdot int
1
0
1199102
119909119909119889119909 + (120582 minus
120578
2+120582120578
2+ 120572119890120582120591)
sdot int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573 +
120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
119890120582119905int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(27)
Proof By definition we see that
1198811(119905) = 119890
2120582119905(119864 (119905) + 119866 (119905)) (28)
where 119864(119905) and 119866(119905) are defined as beforeSo
1 (119905) = 119890
2120582119905(2120582 (119864 (119905) + 119866 (119905)) + (119905) + (119905)) (29)
In what follows we will calculate (119905) and (119905)Using integration by parts and the boundary condition
we have
(119905) = int
1
0
119910119909119909 (119909 119905) 119910119909119909119905 (119909 119905) + 119910119905 (119909 119905) 119910119905119905 (119909 119905) 119889119909
= 119910119909119909 (119909 119905) 119910119905119909 (119909 119905)
1003816100381610038161003816
1
0
minus int
1
0
119910119909119909119909 (119909 119905) 119910119905119909 (119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ int
1
0
2120572119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
= 119910119909119909(119909 119905) 119910
119905119909(119909 119905)
1003816100381610038161003816
1
0minus 119910119909119909119909
(119909 119905) 119910119905(119909 119905)
1003816100381610038161003816
1
0
+ int
1
0
119910119909119909119909119909
(119909 119905) 119910119905(119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909 = minus120573119910
2
119905(1 119905)
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
(119905) = 120578 (int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ int
1
0
119909119910119909(119909 119905) 119910
119905119905(119909 119905) 119889119909)
= 120578(int
1
0
119909119910119905119909 (119909 119905) 119910119905 (119909 119905) 119889119909
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
Journal of Difference Equations 5
minus int
1
0
119909119910119909 (119909 119905) 119910119909119909119909 (119909 119905) 119889119909) = 120578(
1
21199102
119905(1 119905)
minus1
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3
2int
1
0
1199102
119909119909(119909 119905) 119889119909
minus 120573119910119909(1 119905) 119910
119905(1 119905)
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909)
(30)where we have used equalities
int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
=1
21199102
119905(1 119905) minus
1
2int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909119909119909119909
(119909 119905) 119910119909(119909 119905) 119889119909
=3
2int
1
0
1199102
119909119909(119909 119905) 119889119909 + 120573119910
119909(1 119905) 119910
119905(1 119905)
(31)
Summarizing the above all we have
1= 1198902120582119905
[120582int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909] + 119890
2120582119905[minus1205731199102
119905(1 119905)
+120578
21199102
119905(1 119905) minus 120573120578119910119909 (1 119905) 119910119905 (1 119905)
minus120578
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
= 1198902120582119905
[(120582 minus3120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
+ 1198902120582119905
[120578 minus 2120573
21199102
119905(1 119905) minus 120573120578119910
119909(1 119905) 119910
119905(1 119905)]
(32)
Since
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
le1
4int
1
0
1199102
119909119909119889119909 +
1
4int
1
0
1199102
119905(119909 119905) 119889119909
minus 120573120578119910119909(1 119905) 119910
119905(1 119905) le
120573120578
21199102
119909(1 119905) +
120573120578
21199102
119905(1 119905)
le120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909 +
120573120578
21199102
119905(1 119905)
(33)
we have
1le 1198902120582119905
[(120582 minus3120578
2+120582120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909] + 119890
2120582119905 120578 minus 2120573
2
sdot 1199102
119905(1 119905) + 119890
2120582119905[120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+120573120578
21199102
119905(1 119905)]
= 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909] + 119890
2120582119905
sdot120578 minus 2120573 + 120573120578
21199102
119905(1 119905)
(34)
We now estimate the integral terms with time delay ApplyingYoungrsquos and Poincarersquos inequalities we have
int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205751
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
21205751
int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205752
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
81205752
int
1
0
1199102
119909119909(119909 119905) 119889119909
(35)
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Difference Equations 3
minus1
120591ℎ120588(119909 120588) = 120596 (119909 120588)
(11)
with boundary condition
119891 (0) = 1198911015840(0) = 119891
10158401015840(1) = 0
119891101584010158401015840(1) = 120573119892 (1)
ℎ (119909 0) = 119892 (119909)
(12)
By a complex calculation we get the solution
119892 (119909) = 120583 (119909)
ℎ (119909 120588) = 120583 (119909) minus int
120588
0
120591120596 (119909 119904) 119889119904
119891 (119909) = minusint
119909
0
int
119903
0
int
119902
1
int
119901
1
(2120572120583 (119904) minus 2120572int
1
0
120591120596 (119904 119896) 119889119896 minus 120592 (119904)) 119889119904 119889119901 119889119902 119889119903 +1
6120573120583 (1) 119909
3
(13)
Let 119891 119892 ℎ be given as (13) Then we have A119884 = 119865 and119884 = (119891 119892 ℎ) isin 119863(A) The closed operator theorem assertsthat 0 isin 120588(119860) and Aminus1 H rarr 119863(A) is a bounded linearoperator Since 119863(A) sub 119867
4
119864(0 1) times 119867
2(0 1) times 119867
1(0 1) the
Sobolev Embedding Theorem asserts that Aminus1 is a compactoperator on H Hence by the spectral theory of compactoperator 120590(A) consists of all isolated eigenvalues of finitemultiplicity
Theorem 2 Let A and H be defined as before Then Agenerates a 119862
0semigroup on H Hence system (10) is well
posed
Proof For any real 119884 = (119891 119892 ℎ)119879isin D(A) we calculate
⟨A119884 119884⟩H = int
1
0
11989210158401015840(119909) 11989110158401015840(119909) 119889119909
+ int
1
0
119892 (119909) (minus119891119909119909119909119909 (119909) + 2120572ℎ (119909 1)) 119889119909
minus∬
1
0
ℎ120588(119909 120588) ℎ (119909 120588) 119889119909 119889120588
= minus119892 (1) 119891101584010158401015840(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
(ℎ2(119909 1) minus ℎ
2(119909 0)) 119889119909
= minus1205731198922(1) + 2120572int
1
0
ℎ (119909 1) 119892 (119909) 119889119909
minus1
2int
1
0
ℎ2(119909 1) 119889119909 +
1
2int
1
0
1198922(119909) 119889119909
(14)
Since 120573 gt 0 we have
⟨A119884 119884⟩H le minus1
2int
1
0
(ℎ (119909 1) minus 2120572119892 (119909))2119889119909
+1
2int
1
0
(41205722+ 1) 119892
2(119909) 119889119909
le 119872⟨119884 119884⟩H
(15)
where119872 = 21205722+12 which shows thatAminus119872119868 is a dissipative
operator This together with Lemma 1 shows that A minus 119872119868
satisfies the conditions of Lumer-Phillips theorem [22] SoAgenerates a 119862
0semigroup onH
3 Exponential Stability of the System
In this section we consider the exponential stability issue ofsystem (1) based on Lyapunov method
The energy function of system (1) is defined as
119864 (119905) =1
2int
1
0
[1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)] 119889119909 (16)
In what follows we will give some lemmas that are thefoundation of our method
Lemma 3 (see [23]) Let 119864(119905) be a nonnegative function onR+ If there exists a function 119881(119905) and some positive numbers
1198881and 120582 such that the conditions
119881 (119905) gt 1198881119890120582119905119864 (119905) forall119905 ge 0 (17)
(119905) le 0 forall119905 ge 0 (18)
hold then 119864(119905) decays exponentially at rate 120582
In order to construct a function 119881(119905) satisfying theconditions in Lemma 3 we set
119866 (119905) = 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909 (19)
where 120578 is a constant and satisfies 0 lt 120578 lt 2
We can establish an equivalence relation between 119866(119905)
and 119864(119905) via the following Lemma
Lemma 4 Let 119864(119905) and 119866(119905) be defined as before Then thereexist positive constants 119888
2and 1198883such that
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) forall119905 ge 0 (20)
holds
4 Journal of Difference Equations
Proof Let 119910(119909 119905) be the solution of (1) Applying Youngrsquos andPoincarersquos inequalities100381610038161003816100381610038161003816100381610038161003816
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
le120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
11990921199102
119909(119909 119905) 119889119909
le120575
2int
1
0
1199091199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
1199092119889119909int
119909
0
1199102
119909119909(119904 119905) 119889119904
lt120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
8120575int
1
0
1199102
119909119909(119909 119905) 119889119909
forall120575 gt 0 119905 ge 0
(21)
Taking 120575 = 12 we get100381610038161003816100381610038161003816100381610038161003816
120578 int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
lt120578
2119864 (119905) 119905 ge 0 (22)
Since 0 lt 120578 lt 2 we can set 1198881= 1 minus 1205782 and 119888
2= 1 + 1205782
then
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) (23)
The desired inequality follows
Let 120582 gt 0 We define a function 119881(119905) by
119881 (119905) = 1198811(119905) + 119881
2(119905) (24)
where
1198811(119905) = 119890
2120582119905(1
2int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909)
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909
(25)
Noting that 120572 gt 0 according to Lemma 4 we can see that thefollowing result is true
Lemma 5 Let 119881(119905) defined as before Then 119881(119905) satisfiescondition (17) that is
119881 (119905) gt 11988821198902120582119905119864 (119905) 119905 ge 0 (26)
In what follows we calculate (119905) For 1198811(119905) we have the
following result
Lemma 6 Let 1198811(119905) be defined as before and let 119910(119909 119905) be the
solution of (1) Then
1 (119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)
sdot int
1
0
1199102
119909119909119889119909 + (120582 minus
120578
2+120582120578
2+ 120572119890120582120591)
sdot int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573 +
120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
119890120582119905int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(27)
Proof By definition we see that
1198811(119905) = 119890
2120582119905(119864 (119905) + 119866 (119905)) (28)
where 119864(119905) and 119866(119905) are defined as beforeSo
1 (119905) = 119890
2120582119905(2120582 (119864 (119905) + 119866 (119905)) + (119905) + (119905)) (29)
In what follows we will calculate (119905) and (119905)Using integration by parts and the boundary condition
we have
(119905) = int
1
0
119910119909119909 (119909 119905) 119910119909119909119905 (119909 119905) + 119910119905 (119909 119905) 119910119905119905 (119909 119905) 119889119909
= 119910119909119909 (119909 119905) 119910119905119909 (119909 119905)
1003816100381610038161003816
1
0
minus int
1
0
119910119909119909119909 (119909 119905) 119910119905119909 (119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ int
1
0
2120572119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
= 119910119909119909(119909 119905) 119910
119905119909(119909 119905)
1003816100381610038161003816
1
0minus 119910119909119909119909
(119909 119905) 119910119905(119909 119905)
1003816100381610038161003816
1
0
+ int
1
0
119910119909119909119909119909
(119909 119905) 119910119905(119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909 = minus120573119910
2
119905(1 119905)
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
(119905) = 120578 (int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ int
1
0
119909119910119909(119909 119905) 119910
119905119905(119909 119905) 119889119909)
= 120578(int
1
0
119909119910119905119909 (119909 119905) 119910119905 (119909 119905) 119889119909
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
Journal of Difference Equations 5
minus int
1
0
119909119910119909 (119909 119905) 119910119909119909119909 (119909 119905) 119889119909) = 120578(
1
21199102
119905(1 119905)
minus1
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3
2int
1
0
1199102
119909119909(119909 119905) 119889119909
minus 120573119910119909(1 119905) 119910
119905(1 119905)
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909)
(30)where we have used equalities
int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
=1
21199102
119905(1 119905) minus
1
2int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909119909119909119909
(119909 119905) 119910119909(119909 119905) 119889119909
=3
2int
1
0
1199102
119909119909(119909 119905) 119889119909 + 120573119910
119909(1 119905) 119910
119905(1 119905)
(31)
Summarizing the above all we have
1= 1198902120582119905
[120582int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909] + 119890
2120582119905[minus1205731199102
119905(1 119905)
+120578
21199102
119905(1 119905) minus 120573120578119910119909 (1 119905) 119910119905 (1 119905)
minus120578
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
= 1198902120582119905
[(120582 minus3120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
+ 1198902120582119905
[120578 minus 2120573
21199102
119905(1 119905) minus 120573120578119910
119909(1 119905) 119910
119905(1 119905)]
(32)
Since
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
le1
4int
1
0
1199102
119909119909119889119909 +
1
4int
1
0
1199102
119905(119909 119905) 119889119909
minus 120573120578119910119909(1 119905) 119910
119905(1 119905) le
120573120578
21199102
119909(1 119905) +
120573120578
21199102
119905(1 119905)
le120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909 +
120573120578
21199102
119905(1 119905)
(33)
we have
1le 1198902120582119905
[(120582 minus3120578
2+120582120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909] + 119890
2120582119905 120578 minus 2120573
2
sdot 1199102
119905(1 119905) + 119890
2120582119905[120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+120573120578
21199102
119905(1 119905)]
= 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909] + 119890
2120582119905
sdot120578 minus 2120573 + 120573120578
21199102
119905(1 119905)
(34)
We now estimate the integral terms with time delay ApplyingYoungrsquos and Poincarersquos inequalities we have
int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205751
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
21205751
int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205752
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
81205752
int
1
0
1199102
119909119909(119909 119905) 119889119909
(35)
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Difference Equations
Proof Let 119910(119909 119905) be the solution of (1) Applying Youngrsquos andPoincarersquos inequalities100381610038161003816100381610038161003816100381610038161003816
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
le120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
11990921199102
119909(119909 119905) 119889119909
le120575
2int
1
0
1199091199102
119905(119909 119905) 119889119909 +
1
2120575int
1
0
1199092119889119909int
119909
0
1199102
119909119909(119904 119905) 119889119904
lt120575
2int
1
0
1199102
119905(119909 119905) 119889119909 +
1
8120575int
1
0
1199102
119909119909(119909 119905) 119889119909
forall120575 gt 0 119905 ge 0
(21)
Taking 120575 = 12 we get100381610038161003816100381610038161003816100381610038161003816
120578 int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
100381610038161003816100381610038161003816100381610038161003816
lt120578
2119864 (119905) 119905 ge 0 (22)
Since 0 lt 120578 lt 2 we can set 1198881= 1 minus 1205782 and 119888
2= 1 + 1205782
then
1198882119864 (119905) le 119866 (119905) + 119864 (119905) le 119888
3119864 (119905) (23)
The desired inequality follows
Let 120582 gt 0 We define a function 119881(119905) by
119881 (119905) = 1198811(119905) + 119881
2(119905) (24)
where
1198811(119905) = 119890
2120582119905(1
2int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909)
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909
(25)
Noting that 120572 gt 0 according to Lemma 4 we can see that thefollowing result is true
Lemma 5 Let 119881(119905) defined as before Then 119881(119905) satisfiescondition (17) that is
119881 (119905) gt 11988821198902120582119905119864 (119905) 119905 ge 0 (26)
In what follows we calculate (119905) For 1198811(119905) we have the
following result
Lemma 6 Let 1198811(119905) be defined as before and let 119910(119909 119905) be the
solution of (1) Then
1 (119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)
sdot int
1
0
1199102
119909119909119889119909 + (120582 minus
120578
2+120582120578
2+ 120572119890120582120591)
sdot int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573 +
120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
119890120582119905int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(27)
Proof By definition we see that
1198811(119905) = 119890
2120582119905(119864 (119905) + 119866 (119905)) (28)
where 119864(119905) and 119866(119905) are defined as beforeSo
1 (119905) = 119890
2120582119905(2120582 (119864 (119905) + 119866 (119905)) + (119905) + (119905)) (29)
In what follows we will calculate (119905) and (119905)Using integration by parts and the boundary condition
we have
(119905) = int
1
0
119910119909119909 (119909 119905) 119910119909119909119905 (119909 119905) + 119910119905 (119909 119905) 119910119905119905 (119909 119905) 119889119909
= 119910119909119909 (119909 119905) 119910119905119909 (119909 119905)
1003816100381610038161003816
1
0
minus int
1
0
119910119909119909119909 (119909 119905) 119910119905119909 (119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ int
1
0
2120572119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
= 119910119909119909(119909 119905) 119910
119905119909(119909 119905)
1003816100381610038161003816
1
0minus 119910119909119909119909
(119909 119905) 119910119905(119909 119905)
1003816100381610038161003816
1
0
+ int
1
0
119910119909119909119909119909
(119909 119905) 119910119905(119909 119905) 119889119909
minus int
1
0
119910119905(119909 119905) 119910
119909119909119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909 = minus120573119910
2
119905(1 119905)
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
(119905) = 120578 (int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ int
1
0
119909119910119909(119909 119905) 119910
119905119905(119909 119905) 119889119909)
= 120578(int
1
0
119909119910119905119909 (119909 119905) 119910119905 (119909 119905) 119889119909
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
Journal of Difference Equations 5
minus int
1
0
119909119910119909 (119909 119905) 119910119909119909119909 (119909 119905) 119889119909) = 120578(
1
21199102
119905(1 119905)
minus1
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3
2int
1
0
1199102
119909119909(119909 119905) 119889119909
minus 120573119910119909(1 119905) 119910
119905(1 119905)
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909)
(30)where we have used equalities
int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
=1
21199102
119905(1 119905) minus
1
2int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909119909119909119909
(119909 119905) 119910119909(119909 119905) 119889119909
=3
2int
1
0
1199102
119909119909(119909 119905) 119889119909 + 120573119910
119909(1 119905) 119910
119905(1 119905)
(31)
Summarizing the above all we have
1= 1198902120582119905
[120582int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909] + 119890
2120582119905[minus1205731199102
119905(1 119905)
+120578
21199102
119905(1 119905) minus 120573120578119910119909 (1 119905) 119910119905 (1 119905)
minus120578
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
= 1198902120582119905
[(120582 minus3120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
+ 1198902120582119905
[120578 minus 2120573
21199102
119905(1 119905) minus 120573120578119910
119909(1 119905) 119910
119905(1 119905)]
(32)
Since
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
le1
4int
1
0
1199102
119909119909119889119909 +
1
4int
1
0
1199102
119905(119909 119905) 119889119909
minus 120573120578119910119909(1 119905) 119910
119905(1 119905) le
120573120578
21199102
119909(1 119905) +
120573120578
21199102
119905(1 119905)
le120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909 +
120573120578
21199102
119905(1 119905)
(33)
we have
1le 1198902120582119905
[(120582 minus3120578
2+120582120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909] + 119890
2120582119905 120578 minus 2120573
2
sdot 1199102
119905(1 119905) + 119890
2120582119905[120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+120573120578
21199102
119905(1 119905)]
= 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909] + 119890
2120582119905
sdot120578 minus 2120573 + 120573120578
21199102
119905(1 119905)
(34)
We now estimate the integral terms with time delay ApplyingYoungrsquos and Poincarersquos inequalities we have
int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205751
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
21205751
int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205752
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
81205752
int
1
0
1199102
119909119909(119909 119905) 119889119909
(35)
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Difference Equations 5
minus int
1
0
119909119910119909 (119909 119905) 119910119909119909119909 (119909 119905) 119889119909) = 120578(
1
21199102
119905(1 119905)
minus1
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3
2int
1
0
1199102
119909119909(119909 119905) 119889119909
minus 120573119910119909(1 119905) 119910
119905(1 119905)
+ 2120572int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909)
(30)where we have used equalities
int
1
0
119909119910119905119909(119909 119905) 119910
119905(119909 119905) 119889119909
=1
21199102
119905(1 119905) minus
1
2int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909119909119909119909
(119909 119905) 119910119909(119909 119905) 119889119909
=3
2int
1
0
1199102
119909119909(119909 119905) 119889119909 + 120573119910
119909(1 119905) 119910
119905(1 119905)
(31)
Summarizing the above all we have
1= 1198902120582119905
[120582int
1
0
(1199102
119909119909(119909 119905) + 119910
2
119905(119909 119905)) 119889119909
+ 2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909] + 119890
2120582119905[minus1205731199102
119905(1 119905)
+120578
21199102
119905(1 119905) minus 120573120578119910119909 (1 119905) 119910119905 (1 119905)
minus120578
2int
1
0
1199102
119905(119909 119905) 119889119909 minus
3120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
= 1198902120582119905
[(120582 minus3120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120582120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
+ 2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909]
+ 1198902120582119905
[120578 minus 2120573
21199102
119905(1 119905) minus 120573120578119910
119909(1 119905) 119910
119905(1 119905)]
(32)
Since
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905) 119889119909
le1
4int
1
0
1199102
119909119909119889119909 +
1
4int
1
0
1199102
119905(119909 119905) 119889119909
minus 120573120578119910119909(1 119905) 119910
119905(1 119905) le
120573120578
21199102
119909(1 119905) +
120573120578
21199102
119905(1 119905)
le120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909 +
120573120578
21199102
119905(1 119905)
(33)
we have
1le 1198902120582119905
[(120582 minus3120578
2+120582120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909 (119909 119905) 119910119905 (119909 119905 minus 120591) 119889119909] + 119890
2120582119905 120578 minus 2120573
2
sdot 1199102
119905(1 119905) + 119890
2120582119905[120573120578
2int
1
0
1199102
119909119909(119909 119905) 119889119909
+120573120578
21199102
119905(1 119905)]
= 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2)int
1
0
1199102
119909119909(119909 119905) 119889119909
+ (120582 minus120578
2+120582120578
2)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
[2120572int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
+ 2120572120578int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909] + 119890
2120582119905
sdot120578 minus 2120573 + 120573120578
21199102
119905(1 119905)
(34)
We now estimate the integral terms with time delay ApplyingYoungrsquos and Poincarersquos inequalities we have
int
1
0
119910119905(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205751
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
21205751
int
1
0
1199102
119905(119909 119905) 119889119909
int
1
0
119909119910119909(119909 119905) 119910
119905(119909 119905 minus 120591) 119889119909
le1205752
2int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909 +
1
81205752
int
1
0
1199102
119909119909(119909 119905) 119889119909
(35)
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Difference Equations
Thus
1(119905) le 119890
2120582119905[(120582 minus
3120578
2+120582120578
2+120573120578
2+120572120578
41205752
)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+120572
1205751
)int
1
0
1199102
119905(119909 119905) 119889119909] + 119890
2120582119905(minus120573
+120578
2+120573120578
2)1199102
119905(1 119905) + [1205721205751 + 1205721205781205752]
sdot 1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(36)
Taking 1205751= 119890minus120582120591 1205752= 119890minus120582120591
120578 we obtain
1(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 120572119890120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
+ 2120572119890minus120582120591
1198902120582119905
int
1
0
1199102
119905(119909 119905 minus 120591) 119889119909
(37)
The desired inequality follows
Since
1198812 (119905) = 2120572119890
minus120582120591int
1
0
int
119905
119905minus120591
1198902120582(119904+120591)
1199102
119905(119909 119904) 119889119904 119889119909 (38)
we have
2(119905) = 2120572119890
minus120582120591int
1
0
1198902120582(119905+120591)
1199102
119905(119909 119905) 119889119909
minus 2120572119890minus120582120591
int
1
0
11989021205821199051199102
119905(119909 119905 minus 120591) 119889119909
(39)
Employing the estimate we have
(119905) = 1(119905) +
2(119905)
le 1198902120582119905
[(120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782
4119890120582120591)int
1
0
1199102
119909119909119889119909
+ (120582 minus120578
2+120582120578
2+ 3120572119890
120582120591)int
1
0
1199102
119905(119909 119905) 119889119909]
+ 1198902120582119905
(minus120573 +120578
2+120573120578
2)1199102
119905(1 119905)
(40)
Clearly if the parameters 120578 120573 120572 120582 and 120591 are such that theinequalities
120582 minus3120578
2+120582120578
2+120573120578
2+1205721205782119890120582120591
4le 0
120582 minus120578
2+120582120578
2+ 3120572119890
120582120591le 0
minus120573 +120578
2+120573120578
2le 0
(41)
hold then we have (119905) le 0Summarizing discussion above we have the following
result
Theorem 7 Let 119910(119909 119905) be the solution of (1) and let 0 lt 120578 lt 2
and 120582 gt 0 If inequalities (41) hold then the energy function119864(119905) decays exponentially at rate 2120582
We now are in a proposition to study the solvability ofinequalities (41) Noting that 120578 is not a system parameter it isonly amiddle parameterwhich is introduced in themultiplierterm From the third inequality of (41) we see that 120578 and 120573
have a relationship
0 lt 120578 lt2120573
1 + 120573 (42)
Taking 120578 = 2120573(1 + 120573) (41) is equivalent to
120582 minus3120573
1 + 120573+
120582120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890120582120591
(1 + 120573)2le 0
120582 minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572119890
120582120591le 0
(43)
Theorem8 Set 120578 = 2120573(1+120573) If120572 and120573 satisfy the inequality
120572 lt min3120573+ 2 minus 120573
120573
3 (1 + 120573) (44)
then there exists 120582lowast gt 0 such that for 120582 isin (0 120582lowast] inequality
(41) holds true
Proof If (44) holds then
120572 lt3
120573+ 2 minus 120573
120572 lt120573
3 (1 + 120573)
(45)
or equivalently
minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
(46)
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Difference Equations 7
Set
119891 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119909 ge 0
119892 (119910) = 119910 minus120573
1 + 120573+
119910120573
1 + 120573+ 3120572119890
119910120591 119910 ge 0
(47)
Since
119891 (0) = minus3120573
1 + 120573+
1205732
1 + 120573+
1205721205732
(1 + 120573)2lt 0
119892 (0) = minus120573
1 + 120573+
120582120573
1 + 120573+ 3120572 lt 0
119891 (3) = 3 +1205732
1 + 120573+
12057212057321198903120591
(1 + 120573)2gt 0
119892 (1) = 1 + 3120572119890120591gt 0
(48)
there exist 119909lowast isin (0 3) and 119910lowast isin (0 1) such that 119891(119909lowast) = 0
and 119892(119910lowast) = 0Set
119909lowast= min 119909lowast isin (03) 119891 (119909lowast) = 0
119910lowast= min 119910lowast isin (0 1) 119892 (119910lowast) = 0
(49)
and 120582lowast= min119909
lowast 119910lowast Clearly when 120582 isin (0 120582
lowast] we have
119891(120582) le 0 and 119892(120582) le 0 so (43) holds Hence (41) holds true
In what follows we discuss the property of the function
119866 (120573) = min 3120573+ 2 minus 120573
120573
3 (1 + 120573) 120573 ge 0 (50)
We consider equation
3
120573+ 2 minus 120573 =
120573
3 (1 + 120573) (51)
and it is equivalent to
9 + 15120573 + 21205732minus 31205733= 0 (52)
This equation has three real roots 1205731lt 1205732lt 0 lt 2 lt 120573
3lt 3
So we have
119866 (120573) =
120573
3 (1 + 120573) 120573 isin (0 120573
3]
3
120573+ 2 minus 120573 120573 isin [120573
3 3]
3
120573+ 2 minus 120573 lt 0 120573 ge 3
(53)
max120573gt0
119866 (120573) =1205733
3 (1 + 1205733) (54)
0 05 1 15 2 25 3 350
01
02
03
04
120572
120573
120572 =3
120573+ 2 minus 120573 120572 =
120573
3(1 + 120573)
Figure 1 The graph of function 119866 which gives the relationshipbetween 120572 and 120573
31 (120573 120572)-Region of the Exponential Stability According to(52) we determine 120573
3≃ 2818 And according to (53) we draw
the (120573 120572)-regionThe picture of 119866(120573) is given as Figure 1 (120573 120572)-region is
given by
sum(120573 120572) = (120573 120572) | 120572 lt 119866 (120573) 120573 isin (0 3) (55)
Figure 1 gives the graph of function 119866(120573) that givesthe relationship between 120572 and 120573 with which system (1) isexponentially stable From this picture we see that if 120572 islarger we cannot stabilize it by the boundary damping 120572 hasupper bound 120572lowast = 120573
33(1 + 120573
3) ≃ 0246
32 The Best Decay Rate 120582lowast Suppose that 120572 lt 119866(120573) with 120573 isin
(0 3) According to (43) we determine the best decay rate 120582lowastNote that 120582lowast = min119909
lowast 119910lowast isin (0 1) where 119909
lowastand 119910
lowastsolve
the following equation respectively
119909lowastminus
3120573
1 + 120573+
119909lowast120573
1 + 120573+
1205732
1 + 120573+1205721205732119890119909lowast120591
(1 + 120573)2= 0
119910lowastminus
120573
1 + 120573+
119910lowast120573
1 + 120573+ 3120572119890
119910lowast120591= 0
(56)
Firstly let
119901 (119909) = 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
119902 (119909) = 119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591
(57)
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Difference Equations
012 0122 0124 0126 0128 013 0132 0134 0136 0138 014
0
0005
001
0015
y
x
minus0005
minus001
minus0015
058 06 062 064 066 068 07
0
002
004
006
008
y
x
minus002
minus004
minus006
minus008
minus01
Figure 2 120573 = 1 120572 = 01 and 120591 = 01
Obviously 119901(119909) and 119902(119909) both are monotonic functionFor comparing to the two equations of (56) we define thefunction 119891(119909)
119891 (119909) = 119901 (119909) minus 119902 (119909)
= 119909 minus3120573
1 + 120573+
119909120573
1 + 120573+
1205732
1 + 120573+
1205721205732119890119909120591
(1 + 120573)2
minus (119909 minus120573
1 + 120573+
119909120573
1 + 120573+ 3120572119890
119909120591)
=1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] 119890
119909120591
(58)
Since 120573 120572 120591 gt 0 we have 120572[1205732(1 + 120573)2minus 3]119890119909120591
lt 0 When120573 isin (0 2] we get (1205732 minus 2120573)(1 + 120573) lt 0 that is 119891(119909) lt 0Therefore when 120573 isin (0 2] 120582lowast = 119910
lowast
An example is given Assume that 120573 = 1 120572 = 01 120591 = 01
(ie they are constants) We obtain the approximative zeropoints of 119901(119909) and 119902(119909) from Figure 2 That is 119909
lowast≃ 06489
and 119910lowast≃ 01307 Therefore we get the best decay rate 120582lowast =
119910lowast≃ 01307Next we consider 120573 isin (2 120573
3) By researching (58) we can
obtain (1205732minus 2120573)(1 + 120573) gt 0 and [120573
2(1 + 120573)
2minus 3]119890119909120591
le
1205732(1 + 120573)
2minus 3 le 0 We have 119891(119909) le (120573
2minus 2120573)(1 + 120573) +
120572[1205732(1 + 120573)
2minus 3]
Let
119867 =1205732minus 2120573
1 + 120573+ 120572[
1205732
(1 + 120573)2minus 3] lt 0 (59)
We have 120572 gt (2120573minus1205732)(1+120573)(120573
2minus3(1+120573)
2) gt 0 According
to (120573 120572)-region of the exponential stability we obtain 120572 lt
1205733(1 + 120573) After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
lt 120572 lt120573
3 (1 + 120573) (60)
To summarize 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus
3(1 + 120573)2) lt 120572 lt 1205733(1 + 120573) we can get 119891(119909) le 119867 lt 0
Therefore the best decay 120582lowast = 119910lowast
Then we give two examplesAssume that 120573 = 21 isin (2 120573
3] 120572 = 01 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 3 Weget the best decay rate 120582lowast = 119910
lowast
Assume that 120573 = 21 isin (2 1205733] 120572 = 02 isin (00267 02258)
120591 = 01 (ie they are constants) we can obtain Figure 4 Weget the best decay rate 120582lowast = 119910
lowast
Finally we consider 120573 isin (1205733 3) We can also obtain 120572 gt
(2120573 minus 1205732)(1 + 120573)(120573
2minus 3(1 + 120573)
2) gt 0 when 119891(119909) le 119867 lt 0
Here 120572 lt 3120573 + 2 minus 120573 After comparison
(2120573 minus 1205732) (1 + 120573)
1205732 minus 3 (1 + 120573)2
gt3
120573+ 2 minus 120573 (61)
Therefore Not all 119891(119909) lt 0 were always correctSummarizing the above all the best decay 120582lowast is easy to
determine when 120573 isin (0 2) The best decay 120582lowast is not easy todetermine when 120573 isin (2 3) We have two conclusion
(i) 120573 isin (0 2] the best decay 120582lowast = 119910lowast
(ii) 120573 isin (2 1205733] when (2120573 minus 1205732)(1 + 120573)(1205732 minus 3(1 + 120573)2) lt
120572 lt 1205733(1 + 120573) the best decay 120582lowast = 119910lowast
4 Conclusions
In this paper using the Lyapunov functional approach wediscussed the exponential stabilization of an Euler-Bernoullibeam equation with interior delays and boundary dampingDifferent from the earlier papers we added a multiplierterm 119890
2120582119905 to the Lyapunov function so as to transform theexponential stability By solving the inequality equations wegive the exponential stability region of the system
We note that the method used in this paper also canapply to the investigation of the exponential stability of other
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Difference Equations 9
02 0205 021 0215 022 0225 023 0235 024 0245 025
0
001
002
003
004
005
x
y
minus001
minus002
minus003
minus0040305 031 0315 032 0325 033 0335 034 0345 035
0
001
002
003
y
x
minus001
minus002
minus003
minus004
minus005
Figure 3 120573 = 21 120572 = 01 and 120591 = 01
01 015 02 025 03 035 04 045
0
01
02
y
x
minus01
minus02
minus03
0 005 01 015
0
005
01
015
02
y
x
minus005minus01
minus005
minus01
minus015
minus02
minus025
Figure 4 120573 = 21 120572 = 02 and 120591 = 01
model In the future we will study the boundary feedbackcontrol anti-interior time delay for other models
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This research is supported by the Natural Science Founda-tion of China (Grant nos NSFC-61174080 61573252 and61503275)
References
[1] R Datko ldquoTwo questions concerning the boundary control ofcertain elastic systemsrdquo Journal of Differential Equations vol 92no 1 pp 27ndash44 1991
[2] R Datko ldquoTwo examples of ill-posedness with respect to smalltime delays in stabilized elastic systemsrdquo IEEE Transactions onAutomatic Control vol 38 no 1 pp 163ndash166 1993
[3] A Guesmia and S A Messaoudi ldquoGeneral energy decay esti-mates of Timoshenko systemswith frictional versus viscoelasticdampingrdquo Mathematical Methods in the Applied Sciences vol32 no 16 pp 2102ndash2122 2009
[4] S A Messaoudi and B Said-Houari ldquoUniform decay in aTimoshenko-type system with past historyrdquo Journal of Mathe-matical Analysis and Applications vol 360 no 2 pp 459ndash4752009
[5] Q Y Dai and Z F Yang ldquoGlobal existence and exponentialdecay of the solution for a viscoelastic wave equation with adelayrdquo Zeitschrift Angewandte Mathematik Und Physik vol 65no 5 pp 885ndash903 2014
[6] S Nicaise and C Pignotti ldquoStabilization of second-order evolu-tion equations with time delayrdquoMathematics of Control Signalsand Systems vol 26 no 4 pp 563ndash588 2014
[7] S Nicaise and C Pignotti ldquoExponential stability of abstractevolution equations with time delayrdquo Journal of EvolutionEquations vol 15 no 1 pp 107ndash129 2015
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Difference Equations
[8] H Wang and G Q Xu ldquoExponential stabilization of 1-d waveequation with input delayrdquoWSEAS Transactions on Mathemat-ics vol 12 no 10 pp 1001ndash1013 2013
[9] G Xu and HWang ldquoStabilisation of Timoshenko beam systemwith delay in the boundary controlrdquo International Journal ofControl vol 86 no 6 pp 1165ndash1178 2013
[10] X F Liu and G Q Xu ldquoExponential stabilization for Timo-shenko beam with distributed delay in the boundary controlrdquoAbstract and Applied Analysis vol 2013 Article ID 726794 15pages 2013
[11] S Nicaise and C Pignotti ldquoStability and instability results of thewave equation with a delay term in the boundary or internalfeedbacksrdquo SIAM Journal on Control and Optimization vol 45no 5 pp 1561ndash1585 2006
[12] S Nicaise and C Pignotti ldquoStabilization of the wave equationwith boundary or internal distributed delayrdquo Differential andIntegral Equations vol 21 no 9-10 pp 935ndash958 2008
[13] M Krstic and A Smyshlyaev ldquoBackstepping boundary controlfor first-order hyperbolic PDEs and application to systems withactuator and sensor delaysrdquo Systems amp Control Letters vol 57no 9 pp 750ndash758 2008
[14] J Y Park Y H Kang and J A Kim ldquoExistence and exponentialstability for a Euler-Bernoulli beam equation with memoryand boundary output feedback control termrdquo Acta ApplicandaeMathematicae vol 104 no 3 pp 287ndash301 2008
[15] J Liang Y Chen and B-Z Guo ldquoA new boundary controlmethod for beam equation with delayed boundary measure-ment using modified smith predictorsrdquo in Proceedings of the42nd IEEE Conference on Decision and Control vol 1 pp 809ndash814 Maui Hawaii USA 2003
[16] Y F Shang G Q Xu and Y L Chen ldquoStability analysis of Euler-Bernoulli beamwith input delay in the boundary controlrdquoAsianJournal of Control vol 14 no 1 pp 186ndash196 2012
[17] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems ampControl Letters vol 61 no 11 pp 1069ndash1078 2012
[18] Z-J Han and G-Q Xu ldquoOutput-based stabilization of Euler-Bernoulli beam with time-delay in boundary inputrdquo IMAJournal of Mathematical Control and Information vol 31 no 4pp 533ndash550 2013
[19] K Y Yang J J Li and J Zhang ldquoStabilization of an Euler-Bernoulli beam equations with variable cofficients underdelayed boundary output feedbackrdquo Electronic Journal of Dif-ferential Equations vol 75 pp 1ndash14 2015
[20] B-Z Guo and K-Y Yang ldquoDynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observa-tionrdquo Automatica vol 45 no 6 pp 1468ndash1475 2009
[21] F-F Jin and B-Z Guo ldquoLyapunov approach to output feed-back stabilization for the Euler-Bernoulli beam equation withboundary input disturbancerdquo Automatica vol 52 pp 95ndash1022015
[22] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
[23] T Caraballo J Real and L Shaikhet ldquoMethod of Lyapunovfunctionals construction in stability of delay evolution equa-tionsrdquo Journal of Mathematical Analysis and Applications vol334 no 2 pp 1130ndash1145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![On the Exponential Decay of Correlation Functions* · This is analogous to the result [4] that an exponential bound for the decay of the Ursell functions for sufficiently large Λ,](https://static.fdocuments.in/doc/165x107/5f087e397e708231d4224850/on-the-exponential-decay-of-correlation-functions-this-is-analogous-to-the-result.jpg)
