Eur. Transp. Res. Rev. (2014) 6:71–84DOI 10.1007/s12544-013-0106-0

ORIGINAL PAPER

Environmental impact reduction of commercial aircraftaround airports. Less noise and less fuel consumption

Salah Khardi

Received: 9 March 2012 / Accepted: 29 May 2013 / Published online: 9 July 2013© The Author(s) 2013. This article is published with open access at SpringerLink.com

AbstractIntroduction Flight path optimization is designed for min-imizing environmental impacts of aircraft around airportsduring approaches. The main objective of this paper isto develop a model of optimal flight paths taking intoaccount jet noise, fuel consumption, constraints and extremeoperational limits of the aircraft in approach.Materials and methods Optimal control problem combiningaircraft flight paths, noise and fuel consumption is modeledand solved. Outputs, algorithm efficiency and their abil-ity to be interfaced with the in-flight management system,respecting airspace system regulation constraints, are ana-lyzed. Measurements of aircraft noise are carried out aroundSaint-Exupery Lyon International Airport for one year, andINM simulations are performed.Results and conclusion A two-segment approach is con-sidered as an optimal trajectory. Aircraft alignment on therunway axis with a slope between 3◦ and 4.5◦ during the lastapproach segment is obtained and the descent rate is about1060 ft/mn. This particularly characterizes the continuousdescent approach having the potential to reduce noise emis-sion by −4 dB and fuel consumption by −20 % to −10 %during the approach. Because of the suggested trajectory,optimized noise levels are less than the measured and INMvalues given by the empirical trajectory. Optimal trajectoryconsumes less than the standard one; and it can be integratedin the aircraft FMS and the autopilot system. This is one of

S. Khardi (�)Transport and Environment Laboratory, The French Instituteof Science and Technology for Transport,Development and Networks, IFSTTAR,25 avenue F. Mitterrand, 69675 Bron, Francee-mail: salah.khardi@ifsttar.fr

the promising objectives of this research. To conclude, envi-ronmental impacts and fuel consumption are reduced by theuse of aircraft trajectory optimization during arrivals.

Keywords Environment · Impacts · Aircraft · Flight path ·Noise reduction · Fuel consumption

1 Introduction

The airspace system is becoming increasingly congested asthe number of aircraft operations grows to meet passengerand goods demands. The bulk of traffic contributes to airporttraffic saturation. Due to this increase, populations livingnear airports, as well as the environment, are impacted bycommercial aircraft.

Aircraft noise is considered to be one of the most signif-icant environmental concerns in the local communities ofmodern cities, affecting people living near airports. This issupported by evidence that aircraft noise exposure is associ-ated with reduced well-being, lower self-reported quality oflife and higher levels of self-reported stress, anxiety, depres-sion and psychological morbidity. Significant work has beendone in the area of noise effects reflecting different aspectsof annoyance and human health considerations. This is acritical issue that affects the sustainability of commercialaviation. Different solutions have been attempted to con-trol aircraft noise at airports. Nevertheless, noise levels inthe vicinity of airports, in particular under the take-off andlanding flight paths remain high and disrupt the quality oflife of local residents. This is considered today to be oneof the most significant environmental concerns affectingpopulation and the environment [1].

The historical trend in aircraft noise has shown a reduc-tion of approximately 20 dB since the 1960s largely due to

72 Eur. Transp. Res. Rev. (2014) 6:71–84

the adoption of high bypass turbofans and more effectivelining materials. Reductions since the mid-eighties have notbeen as dramatic. The point seems to have been reachedwhere future improvements through technological advanceswill be possible only by significantly trading off operatingcosts for environmental performance.

In addition, a number of major European airports havereached their environmental capacity before having madefull use of their runway and terminal infrastructures. Thus,the significant environmental challenges of the AdvisoryCouncil for Aeronautics Research in Europe have beensub-divided into four goals:

– To reduce fuel consumption and CO2 emissions by50 %;

– To reduce perceived external noise by 50 %;– To reduce NOx by 80 %;– To make substantial progress in reducing the environ-

mental impact of the manufacture, maintenance anddisposal of aircraft and related products.

Contributors to noise reduction should encompass tech-nology related elements such as the Quiet Aircraft andRotorcraft of the future as well as further actions aimed atestablishing efficient Environmental Practices by way ofNoise Abatement Procedures and Management of NoiseImpact.

The control of noise around airports is a complex matterbecause it depends on many different factors, and manydifferent issues are involved. Significant research is cur-rently being undertaken with the goal of reducing aircraftnoise. Existing projects have already identified a number ofpromising procedures and methods that could be applied toachieve an harmonized approach within the EU and furthersteps should go a step further, by developing enabling tech-nologies, tools and methods to ensure the safe, efficient andeconomic operation of new noise abatement procedures.The suggested research activities for aircraft should coverboth:

– The definition of noise abatement flight proceduresthrough flight test demonstrations, taking account of thesafety and of operational constraints.

– The development of pilot aids to ease the differentoperations through specific control laws using specificguidance system (as for instance GPS or D-GPS).

– The development of real time noise footprint assess-ment to be used for in-flight demonstration and forpiloted simulation to elaborate control strategies.

These activities generally fall into one of the three cate-gories: noise source modifications (e.g. retrofitting enginenacelles with sound absorption material), land use regula-tions (e.g. zoning land near an airport as nonresidential), andoperational controls (e.g. power cutbacks during operations;

this is considered impractical because the pilot workloadsand flight safety decrease). Modification of the ends of flapsections and other aerodynamic treatment will have somesuccess; but there is no great optimism. It is necessaryto examine the potential for modified approach proce-dures that can benefit both airlines and the communities.The International Civil Aviation Organization (ICAO) hasrecognized the potential benefits of using advanced flightguidance technology to reduce the impact of aircraft noise[2, 23, 26, 42]. ICAO charged its Noise Abatement Oper-ating Measures Subgroup (NAOMSG) with the followingtasks:

1. Describe effective existing noise abatement operationalprocedures and strategies.

2. Evaluate the critical components of aircraft flight pro-cedures that can minimize source noise emissions andcommunity exposure.

3. Identify emerging and future airport systems technolo-gies in the fields of flight management, ATC andairport capacity which could also serve to minimizecommunity noise exposure.

4. Conceive new operating procedures to reduce commu-nity noise exposure taking into account the emergingand future technologies identified in 3.

It should be remembered that the assessment of theappropriate noise abatement procedure to use needs toconsider several coupled factors:

– aircraft performance and trajectory;– noise generated by the aircraft;– population distribution and density;– flight safety and pilot acceptance;– guidance and navigation requirements;– local atmospheric conditions.

Advanced flight guidance technologies, which are cur-rently in use, such as Area Navigation (RNAV) utilizingthe Global Positioning System (GPS), offer the potentialto reduce the impact of aircraft noise in communities sur-rounding airports by enabling more flexible approach anddeparture procedures that reduce noise exposure to the mostsensitive areas.

A balanced approach to environmental protection hasbeen developed under ICAO and it includes protectionfrom excessive aircraft noise first. There are several pro-cedures that have been implemented in aircraft operationworld-wide:

1) low-noise take-off and approach flight procedures;2) optimal distribution of the aircraft among the routes;3) flight route optimization in the airport vicinity. In com-

bination, the aircraft, airport, and the community forma closed system. If flight procedures are the means of

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operating that system, then the noise abatement pro-cedure represents a way of operating the system withlower noise impact.

The Committee on Aviation Environmental Protec-tion (CAEP) presented the following described BalancedApproach on aircraft noise management around airports [2,17, 26, 41].

Concept of a balanced approach An aircraft is a majorinvestment, with a useful economic life of 25 years or more.Operation of an aircraft includes airframe and engine per-formance. The performance of an aircraft must address theoverriding issue of safety, as well as mission or performanceefficiencies, economics, and environmental objectives.

Across the various organizations, the EU, the UnitedStates, and ICAO the fundamental approach to analysisand management or control of aircraft noise is similar.All recommend a “balanced” approach that includes atleast four ingredients: noise control at the source (the air-craft), land-use planning around airports, noise abatementoperating procedures for aircraft, and restrictions on oper-ations. In addition to these four categories, in the UnitedStates, the FAA includes airport layout and ICAO rec-ognizes noise charges. To pursue the balanced approach,specific tools are required. First, metrics or indicators ofnoise must be identified, noise effects or impacts in rela-tion to the indicators need to be defined, and a methodfor computing the values of the metrics in communitiesaround airports is required. The ICAO Assembly endorsedthe concept of a balanced approach to aircraft noise man-agement. Within ICAO, CAEP developed the requisiteguidance material. Four distinct elements have to be con-sidered and analyzed: Airspace, airfield, terminal, groundaccess.

Most of the world major airports have operational con-straints or capacity limits based upon noise. But the futurepotential growths of air traffic imply that emission sourcesin the future will increase in importance. The study of inte-grated airport impact shows that it is necessary to introducethe concept of airport traffic (operational) capacity accord-ing to environmental safety conditions. Evaluation of anairport impact on surrounding environment could be real-ized by defining environmental capacity of an airport. Itmeans reduction of an airport’s capacity so as to ensure thatairport environmental performances comply with the envi-ronmental rules. Operational capacity of an airport can bemeasured as the number of runway-taxiways slots, the ter-minal capacity or capacity of the apron areas. It is limitedonly by means of flight safety. The economic capacity ofan airport can be measured as the maximum number ofpassengers or aircraft, which can be accommodated on aparticular day with a given amount of infrastructure under

given economic conditions. In a short term, airport ser-vice load during peak and off-peak period determine theseconditions. In long term, the availability of investmentsfor airport expansion principally determines the economicconditions.

The impacts of the airport operation upon the localenvironment are a major issue, which will affect both thecapacity and the potential for future growth. This conceptof environmental capacity as it applies to airports can beapproached in at least two ways: the first is that an airportoperational capacity is less than the total sum of the indi-vidual environmental mitigation measure already in placeat that airport. The second is or could ever lead to an envi-ronmentally optimal solution. It is necessary to identify andseparate short term concerns which mainly affect quality oflife (e.g. aircraft noise) from long term issues which mainlyaffect the assimilative capacity of the environment to copewith what we are throwing at it (e.g. pollution and globalwarming). In addition, it is necessary to assess the viabil-ity of the environmental mitigation measures that are inthe airport territory and in the vicinity. For example, manymajor airports have long-established night flight restrictionswhose aim has been to protect local communities fromexcessive exposure from aircraft noise. From an environ-mental capacity perspective, such restrictions may be seenas short-run, quality of life issue and a successful mitiga-tion measure but with potentially more serious long termenvironmental consequences.

Thus, CAEP identified noise problem and discussed themeasures for its reduction and control. Among four ele-ments of a Balanced Approach, current investigations dealwith Noise Abatement Procedures (NAP), which accord-ing to ICAO’s policies, enable the reduction of noise duringaircraft operations. New flight path development is a solu-tion which should contribute to a decrease of aircraft noiseannoyance. It will also meet objectives 2020 of the AdvisoryCouncil for Aeronautics Research in Europe (ACARE).CAEP, OACI and ACARE reported that flight path opti-mization can provide a sizable decrease in noise impactdepending not only on the population distribution, but onthe types and the numbers of aircraft operations.

This environmental problem can only be solved withinthe framework of a balanced global vision for a sustain-able air transport involving new technology engines andfuselages, breakthrough technologies, the design of newprocedures and flight paths, airspace management, newregulation rules and certification. Thus, technological devel-opments, airspace management, operational improvementand system efficiency should be considered as environ-mental innovations. There is no justification that air trans-port will not continue to progress without decreasing itsenvironmental impacts [2]. The applied procedures arenot optimized but are generic in nature. New flight path

74 Eur. Transp. Res. Rev. (2014) 6:71–84

optimization, associated with new aircraft design andengines, is a solution which should contribute to a decreasein aircraft annoyances. Noise abatement procedures are con-sidered as a necessary measure for a balanced approach ofnoise control around airports and for fuel consumption sav-ing. Any system, which defines the correct features of theoptimized flight paths for aircraft in specific conditions,would be useful for environmental impact control. Develop-ments cannot be carried out without improvement modeling.This consists of developing efficient processing tools whichallow in-flight diagnosis and control in real-time taking intoaccount the FMS (flight management system) functionali-ties and the AMS (airspace management system) updates.Flight path optimization is an innovative solution in theshort run, making a significant contribution to the reductionof commercial aircraft impact on the environment possible.

This paper presents a dynamic method providing optimalflight paths which minimize aircraft impacts and fuel con-sumption. This is an optimal control problem to be solved.The main features of the suggested method are its effective-ness and its resolution speed. It shows a significant potentialof its own integration in the avionic systems. That is whyin this paper we have solved the flight path optimizationproblem with the aim of confirming its advantages, reli-ability and features. We have suggested an optimizationmethod for solving a model governed by an ODE system[3, 4] providing the best flight path suitable for noise andfuel consumption reduction. The cost function of this modeldescribes aircraft noise and fuel consumption [5, 6]. TheODE depends on the flight dynamics of the aircraft, andconsiders flight safety and stability requirements. Numeri-cal methods which solve the ODE fall into several categories[7–11] which depend on the case study. Thus, it is neces-sary to choose, to improve or develop a new method. Inthis context, this paper gives numerical considerations andalgorithms for solving the control problem stressing thecomputing times with the aim of finding the best aircraftapproach able to reduce noise and favoring economizationof fuel consumption. The applied approach has been usedto reduce the (OCP) to a finite-dimensional nonlinear pro-gram which is solved by a standard nonlinear programmingsolver.

Optimality conditions, given by Pontryagin’s principle,have been discretized. A combination of the AMPL model(A Modeling Language for Mathematical Programming)[12] and an NLP solver [13–15] has been performed forcalculations. In-depth details have been described in previ-ous papers [16–19]. Technically, we analyze the processingoutputs and algorithm efficiency and their ability to beinterfaced with the in-flight management system respect-ing airspace system regulation constraints. This integrationcould compensate both the growth in air traffic and theencroachment of airport-neighboring communities.

This paper presents an introduction giving the optimalcontrol problem and resolution, numerical results followedby an experimental measurements analysis and a con-clusion. Measurements of aircraft noise, recorded underthe flight path close to Saint-Exupery Lyon InternationalAirport, are given. To validate the optimization method,the measured noise levels were compared to noise valuesobtained by INM for standard trajectories and by optimiza-tion method of flight path.

2 Optimal control problem and resolution

We present in this section a summary of the optimal controlproblem to be solved which is described in previous papers[16–19]. The system of differential equations commonlyemployed in aircraft trajectory analysis is the following six-dimensional system (ED) derived at the center of mass ofthe aircraft [16, 20]:

(ED1)

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

V = g

(T cosα −D

mg− sin γ

)

γ = 1

mV((T sinα + L) cosμ−mg cos γ )

χ = (T sinα + L) sinμ

mV cos γ

(ED2)

⎧⎨

⎩

x = V cos γ cosχy = V cos γ sinχh = V sin γ

where V, γ, χ, α and μ are respectively the speed, the angleof descent, the yaw angle, the angle of attack and the rollangle. (x, y, h) is the position of the aircraft. The variablesT ,D,L,m and g are respectively the engine thrust, thedrag force, the lift force, the aircraft mass and the aircraftweight acceleration given in previous papers [16, 19–21].(ED) can be written in the following matrix form z(t) =f (z(t), u(t)), where:

z : [t0, tf ] �→ R6

t �→ z(t) = [V (t), γ (t), χ(t), x(t), y(t), h(t)]are the state variables

u : [t0, tf ] �→ R3

t �→ u(t) = [α(t), δx(t), μ(t)] are the control variablesand t0 and tf are the initial and final times.

Along the flight path, we have to respect parameter limitsrelated to the flight safety and the operational modes of theaircraft. Aircraft modeling continues to meet the increaseddemands associated with aviation and airport expansion.Aircraft noise footprints are commonly used for forecast-ing the impact of new developments, quantifying the noise

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trends around airports and evaluating new tools. Thus, air-craft models have become more sophisticated and theirvalidation complex. A number of them are entirely basedon empirical data. Because of this complexity, such modelsare not characterized by a given analytical form describingnoise at reception points on the ground. This paper uses thebasic principles of aircraft noise modeling. The cost func-tion may be chosen as any of the usual aircraft noise indices,which describes the effective noise level of the aircraft noiseevent [22, 23]. This study is limited to minimize the maxi-mum noise level using a semi-empirical model of jet noise[5, 6, 24–26]. The cost function is expressed in the followingform:

J : C1([t0, tf ], R6

) × C1([t0, tf ],R4

) �→ R

J(X(t), U(t))=∫ tf

t0

(�(X(t),U(t))+φ(X(t), tf − t0))dt

where J is the criterion which optimizes noise levels com-pleted by the cost describing fuel consumption φ. The costfunction, related to maximum emitted noise and fuel con-sumption function, can be written in the following integralform:

J : C1([t0, tf ],R6

) × C1([t0, tf ],R3

) �→ R

J (z(t), u(t)) =∫ tf

t0

(�(z(t), u(t))+ φ(z(t), (tf − t0)))dt

where J is the criterion to be optimized. Finding an optimaltrajectory can be stated as an optimal control problem asfollows (t0 = 0):

(OCP)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

min J (z, u)=∫ tf

0(�(z(t), u(t))+φ(z(t), tf ))dt

z(t) = f (z(t), u(t)),∀t ∈ [0, tf ]zI1(0) = c1, zI2(tf ) = c2

a ≤ C(z(t), u(t)) ≤ b

where J : Rn+m → R, f : Rn+m → Rn and C : Rn+m →Rq correspond respectively to the cost function, thedynamic of the problem having a unique state trajectoryand the constraints. The second equation giving the trajec-tory is a nonlinear system with states in Rn. In this paper,limit conditions are specified and their values are given orkept free. tf is also fixed. Different methods for solving theoptimal control problem exist in the open literature. Thereare no practical theoretical limitations to using those meth-ods that cannot be guaranteed to provide a global solution.We assume that the problem has an optimal solution withan optimal cost. In this paper, we have applied an indirectmethod [27–30], based on Pontryagin’s principle [33–36]assessing the solutions. Inequality constraints are carried outby Pontryagin’s maximum principle. We transformed the(OCP) into a new unconstrained (OCP) formulation thatcan be numerically solved. The new unconstrained (OCP)

is obtained by having the same system dimension. To solve(OCP) problems, many methods exist in the open litera-ture [31–36]. In this paper, we have used the optimal controltechnique developed in the references [16–19] which wehave summarized. In depth calculation, details are given inthe quoted papers. We stated that:

where λ, μ are the multiplicators associated to the con-straints and p is the costate vector [37]. We describe theoptimality conditions (OC) for the (OCP) as:

The interior point method [7, 10, 11], discretizing theoptimality conditions of the (OC), is used. (OC) is trans-formed into a sequence of problems where the solutionof optimality conditions is a solution of the (OC) prob-lem: Euler scheme discretization combined with Newtonmethod resolution [11, 38] are used. The annex givesdetails of:

• the perturbation of the two last equation of (OC) by apositive parameter ε corresponding to the complemen-tary conditions,

• the optimality conditions, their discretization, and howto solve the discretized problem.

3 Numerical results

We consider an aircraft landing by fixing initial and finalflight conditions as follow:

0◦ ≤ α ≤ +20◦−10◦ ≤ χ ≤ +10◦−5◦ ≤ μ ≤ +5◦−10◦ ≤ φ ≤ +20◦−60 km ≤ x(t) ≤ 0 km−10 km ≤ y(t) ≤ +10 km0 ≤ t ≤ +10min

0◦ ≤ γ ≤ γmax = f ree

85m/s ≤ V (t) ≤ Vmax = f ree

0.2 ≤ δx(t) ≤ δxmax = f ree

125000 kg ≤ m(t) ≤ mf = f ree

3500m ≤ h(t) ≤ hf = f ree

76 Eur. Transp. Res. Rev. (2014) 6:71–84

Vmin = 85m/s represents the aircraft stall velocity. Someof these parameters are kept free. Once the processingsteps and calculation efficiency are confirmed, their limitvalues are found and given. The data used in this optimiza-tion model are taken from an Airbus A300 of which thespecifications are as follow:

• Type : Airbus A300 − 600• Powerplants : Two 262.4 kN GE CF6–80C2 or

2 × 275 kN• Weights : max TOF 165900 kg. Operating empty

90965 kg• Wing span 44.84 m. Wing area 260 m2

• Length 53.60 m. Height 16.54 m• Max speed : Mach 0.84• Fuel max capacity : 62000 l

The three-dimensional analysis is useful in enhancing thereliability of the optimization model which it could beapplied in automatic detection of aircraft noise. We considerR the distance aircraft-observer as:

R = (x − xobs)2 + (y − yobs)

2 + h2

where (xobs, yobs, 0) were the coordinates of the observer onthe ground under the flight path. They were located between10 km and the touchdown point under the flight path. OCPwas discretized along its state z = (V , γ, χ, x, y, h) andcontrol u = (α, δx, μ) variables. The discretization param-eter was N = 100 points because of the solution stability.To solve the Nε and NLP problems, we used the AMPLmodel [12, 17] and a SNOPT solver. They were chosen

after numerous comparisons among other standard solversavailable on the NEOS optimization platform. We used thecall-by-need mechanism which memorized automaticallythe result of the cost function in order to speed up call-by-name evaluation. This method considers a sequence of Nε

problems (tending ε to zero). The problem Nε is initializedby centering the state and the control. Then, we initialize theLagrange multiplicators as follows:

λ = ε(C(z, u)− a)−1, μ = ε(b − C(z, u))−1

For the implementation of the penalty parameter ε andcomputation, we used the following strategy [39]:

εk+1 = εk/a, a > 1

For each iteration of the interior point method, the algorithmfound an optimal solution. Varying ε from 1 to 1.28×10−5,the feasible error increases from 5.6× 10−12 to 2.1× 10−7,and the optimality error on the solution varies from 10−10

to 3.8 × 10−9. The processing time is equal to 5.27 sec(5.836 CPU). Despite feasibility errors and the time process-ing duration, this method provides the optimal trajectory andits flight parameters. The solution trajectory (the optimalone), the Mach number and the aircraft finesse are shown inFig. 1.

Optimization processing confirms a stabilization of theflight. The altitude h decreases with three gradual slopes.We observe a two-segment approach with an alignment onthe runway axis with a slope of 3◦ during the last approachsegment. Angle of descent is stable as recommended byICAO and aircraft certification [40–42] in favor of ourmethod. The flight rate descent is about 1060 ft/mn which

Fig. 1 Optimal flight path,Mach number and aircraftfinesse variation during theapproach phase

50 100 150 200 250 300 350 400 450 500 550

1000

2000

3000

4000

Alti

tude

(m

)

50 100 150 200 250 300 350 400 450 500 5500.2

0.4

0.6

Mac

h N

umbe

r

50 100 150 200 250 300 350 400 450 500 5500

1

2

Time (s)

Airc

raft

fines

se

Eur. Transp. Res. Rev. (2014) 6:71–84 77

is close to the one recommended by ICAO and practicedby the airline companies (1000 ft/mn). The two segmentdurations obtained are respectively 39 sec at the altitude3150 m and 39 sec at 2673 m. This method, characterizingthe continuous descent approach CDA, can be consideredefficient. It should be remembered that the later can notbe the fastest and shortest CDA. Nevertheless, it has thepotential to reduce noise emission and fuel consumptionduring the approach. To conclude, the obtained optimal tra-jectory could be accepted into the airline community. Thesoft two-segment approach puts the aircraft in an appro-priate envelope with margins for wind uncertainties anderrors. There is no question of vortex separation and prob-lems of intercepting a false glide-slope, given that it mustbe intercepted from above. With autopilot or flight direc-tor coupling, this approach would be acceptable for use inregular air carrier service. Aircraft speed, or Mach num-ber, decreases during the first 70 s, remains stable duringalmost 460 s (which corresponds to the whole period ofthe approach), and decreases during the last 70 s of theapproach. This speed behavior is suitable for the continuousdescent approach accompanied by noise and fuel consump-tion reduction. The aircraft finesse is bang-bang between itsbounds (0.5 and 2). At a constant speed, when the altitudedecreases the thrust increases. It increases during the sec-ond flight period. Additionally, in spite of the fact that themaximum input values of the aircraft speed, the throttle set-ting and altitudes are kept free, the method is fast, errors areweak, and the provided calculations are exact with a correctconvergence. If the guidance is satisfactory and the visualconditions are met, then CDA is considered as a managedapproach. It allows avoidance of flight difficulties, in par-ticular when the speed falls below the speed target -5 knotsor rises above the speed target +10 knots, the pitch altitudebecomes lower than −5◦ or greater than 12◦ nose up, thebank angle becomes greater than 7◦, and when the descentrate varies suddenly around 1000 ft/min. CDA can help theflight crew to make timely and correct thrust settings, andapproach path corrections if necessary. It could consider-ably reduce the go-arounds: pilot workload, fuel and timeof flight reduction. CDA is appropriate for the aircraft sta-bilization in the lower part of 1000 ft with the right thrust.Obviously when the flight is not stabilized the go-aroundbecomes a necessity.

Figure 2 presents noise levels under the flight path versusthe approach distance between 10 km and the touchdownpoint on the runway. This figure shows the evolution ofnoise levels given by INM under the flight path (INM curve)[16, 17] for standard trajectories in accordance with ICAOspecifications [22, 40–42], and by the method described inthis paper. During approach, noise levels increase becausethe aircraft altitude decreases. The tendency curve of the twomethods (INM and optimization) is similar. By comparison,

−10000 −8000 −6000 −4000 −2000 060

65

70

75

80

85

90

95

100

105

110

115

Distance (m)

Noi

se le

vels

(dB

)

Standard Flight Path (INM)

Optimal Flight Path

Fig. 2 Noise levels under the flight path (standard and optimal) vs.approach distance

noise levels obtained by the optimization method are lowerthan those obtained by INM. This method can be consideredthe best in terms of modeling and calculation. It is power-ful and efficient. In spite of the absence of all noise modelsof different aircraft sources, the obtained values are close toexperimental measurements, in particular at the certificationpoint on the ground at 2 km under the flight path.

On the one hand, we can confirm a decrease in noise lev-els using the optimized flight path. At 2 km under the flightpath, the noise level calculated by INM is equal to 93 dBfor standard trajectories and the optimized level is 89 dBfor the optimal trajectory. The mean noise level obtained byINM is 85 dB and that obtained for an optimal trajectoryis 82.5 dB. It should be noted that these calculations havetaken into account the only available model of jet noise. Fur-ther research is needed including all aircraft noise sourceswhen they become available. A 4 dB reduction is obtainedin favor of this method compared to INM calculationsat the certification point. On the other hand, comparisonbetween noise levels corresponding to the standard trajec-tory obtained by INM, and the optimal trajectory given inthis paper, provides changes due to the altitude of approach.Those changes are respectively equal to 4.3 % and 4.5 ofJINM−JOpt

JINMand

JINM−JOpt

JOpt.

In addition, we carried out experiments under the aircraftflight path where we recorded noise levels during one year.Comparisons between calculations and measurements at thecertification point (2 km) under the flight path have beenperformed. The aim of the following work is to validate, byINM, the calculations undertaken.

The measurements of noise, generated by aircraft atapproach, were carried out under the flight path close to

78 Eur. Transp. Res. Rev. (2014) 6:71–84

Lyon Saint-Exupery International Airport (France) for oneyear according to the annex 16 of the ICAO convention.Noise signals were recorded on line. Locations for recordingaircraft noise in flight are surrounded by flat terrain havingno excessive sound absorption characteristics (grass fieldscut). No obstructions exist which could influence the soundfield from the aircraft within a conical space above the pointon the ground vertically below the microphone. The conebeing defined by an axis is normal to the ground and is half-angle (80◦) from this axis. The type of aircraft was recorded;it has been collected in real-time for each flight. In thispaper, we present data recorded under flight path at a lateraldistance of 2 km to the touchdown point. This measurementpoint corresponds to the certification point of the aircraft.The stability of atmospheric conditions was checked andtimetabled. The following numbers show their fluctuationsin the intervals where stability criteria are met during mea-surements. We summarize below meteorological parametersprovided by Meteo France:

– Wind speed (m/s): 1 − 3– Average temperature (◦C): 15 − 35– Cloudiness (octas): 0 − 2– Humidity (%): 35 − 50– Global radiation (J/cm3): 240 − 290

A SIP 95 sound level meter was used to record acousticdata. The measurement system is inspected every two yearsand approved by the French National Laboratory for testingin accordance with international standards. The microphoneis positioned at 4 m above the ground to comply withthe requirement of the free field condition. The ground isflat and consists of grass shorter without brush, wood orobstacles. Two calibrations are performed every day. Thefree-field sensitivity level of the microphone and preampli-fier in the reference direction, at frequencies over at leastthe range of one-third-octave nominal mid-band frequen-cies from 50 Hz to 10 kHz inclusive, is within 1.0 dBof that at the calibration check frequency, and within 2.0dB for nominal mid-band frequencies of 6.3 kHz, 8 kHzand 10 kHz. The output of the analysis system consistsof one-third octave band sound pressure levels as a func-tion of time, obtained by processing the noise signals withthe following characteristics: a set of 24 one-third octaveband filters [50 Hz–10 kHz]; response and averaging prop-erties in which the output from any one-third octave filterband is squared, averaged and displayed or stored as time-averaged sound pressure levels; the interval between succes-sive sound pressure level samples is 500 ms 5 ms for spectralanalysis with or without slow time-weighting. Ambientnoise, including both an acoustical background and elec-trical noise of the measurement system was recorded for10 minutes a day with the system gain set at the lev-els used for the aircraft noise measurements. The recorded

aircraft noise data is acceptable according to internationalstandards. The exclusion criteria of the recorded data are:days on which a strike took place and special weatherconditions (gusty winds, stormy rainfall, atmospheric turbu-lence, etc.). According to the measurement specifications,we identified and retained 15460 turbojet aircraft approach-ing the airport in the same conditions representing 84.5 %(+20 T) of the air traffic (15 % of the air traffic repre-sents propeller aircraft (3–9 T and +20 T) and 0.5 % others(−3 T and 3–9 T)). In this paper, we analyzed the measuredmaximum noise levels. This is because the model describedin the previous sections provide the maximum noisevalues.

Figure 3 presents experimental noise levels obtained atthe certification point for our type of aircraft: among the410 aircraft movements recorded per day, only one aircraftcorresponding to the one chosen for this study landed inthe Lyon Saint-Exupery International Airport corresponded(A300 type). Maximum noise levels of 365 aircraft areshown in the figure (one year measurements). Maximumnoise levels vary from 91.9 to 100.7. These variationsdepend certainly on the atmospheric conditions, the aircraftloads or the effective weight in approach (unavailable infor-mation), and the type of the procedure initiated by the pilots.The average experimental value is 95.7 dB whereas themean levels obtained by INM and optimization processingat 2 km under the flight path are respectively of 93.7 dB et89.3 dB.

On the one hand, the value obtained by INM, at 2 kmunder the flight path is in the interval of the observed val-ues. On the other hand, the optimal value is largely belowthe measured and INM values. Thus, the optimal flight path

50 100 150 200 250 300 35092

93

94

95

96

97

98

99

100

Day

Exp

erim

enta

l noi

se le

vels

(dB

)

Fig. 3 Maximum noise levels measured at 2 km under the flight pathfor one year

Eur. Transp. Res. Rev. (2014) 6:71–84 79

favors a noise level reduction between −11.7 dB and −3dB. This significant noise reduction could be regarded asover-estimated because the other models of fuselage (aero-dynamic) are not introduced in the optimization model.Nevertheless, levels obtained by the optimization and mod-eling processes are promising because the calculated valuesare validated by the experiment at the certification pointof the approach. Because the engine performance couldbe described in several ways, three calculation methods offuel consumption have been carried out using models byHouacine et al. formulation developed by Benson [17, 44],Mattingly [21] and Roux [43] respectively. We have cal-culated the most useful parameter of the engine which isrepresented by the fuel consumption. For turbojets, it isusually expressed as the specific fuel consumption SFC.It is defined as the weight of the fuel burned per timeunit (kg/second in SI units). Using the optimization methodgiven in this paper, we have also calculated the ratio ofaircraft thrust on the aircraft static thrust versus speed ofturbojet and by varying bypass ratio (BPR is defined asthe ratio between the mass flow rate of input air to themass flow rate passing through the engine core which isinvolved in combustion to produce and to increase mechani-cal energy) for showing benefits of optimization in terms ofthrust reduction during the landing. We have used the Mat-tingly and Roux plan model of SFC (PIM curve in Fig. 4)and Houacine et al. (non optimal curve shown in Fig. 4).In this paper, calculation of fuel consumption is introducedas a second term of the objective function . We have usedthe instantaneous fuel consumption representing the fuelburnt during the approach period. Because the initial aircraftmass mt0 is assumed constant and because the weight of theaircraft at the moment t is (mt0 − mt), we can write the

100 200 300 400 500

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Time (s)

Fue

l con

sum

ptio

n (k

g/s)

Non optimal

Optimal

PlM

Fig. 4 Fuel consumption (kg/s) during the approach for three differentmodels

second term of the objective function by the followingexpression:

min{φ(ztf , tf )} = min(−mtf )

where (mt0 −mt) is the final aircraft mass. Figure 4 presentsa comparison result of the fuel consumption calculatedusing the three methods described.

The three used models confirmed a decrease of the fuelconsumption during the approach with a variable stage.Regarding the model of the plan by Mattingly and Roux,fuel consumption decreased with sudden changes, in par-ticular during the beginning and the end of the approach.FC is underestimated. The two other models have no sud-den change in the FC evolution. The Benson model seemssuitable even if it tends to over-estimate. The model ofoptimization provided fuel consumption values between thetwo model limits. What can be retained is the fact that theoptimal trajectory would consume less than the standard tra-jectory. In addition, comparison between models provided10 % and 20 % of

FCNonoptimal−FCOpt

JNonoptimaland

FCOpt−FCPIM

JOpt.

This fuel consumption could be in favor of optimal flightpath. Models describing engine parameters during flightoperations in different conditions are not often published inthe open literature.

Indeed, in-flight engine database is not provided byengine manufacturers or airline companies. Data on FCand thrust versus aircraft Mach, altitude, throttle settingsand power extraction can be correctly modeled because thephysics behind it is well established. The output power ofthe turbojet is quite constant with changes in speed, whereasturbojet provides a constant thrust with speed. Modern air-craft are equipped with engines where the net thrust isreduced with the increase of the speed and the altitude.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.4

0.5

0.6

0.7

0.8

0.9

1

Aircraft Mach Number

Rat

io o

f airc

raft

thru

st /

airc

raft

stat

ic th

rust

BPR5−6

BPR4

BPR3

BPR2

BPR1

BPR0

BPR0−0

Fig. 5 Thrust behavior versus aircraft Mach number with bypass ratiovariation

80 Eur. Transp. Res. Rev. (2014) 6:71–84

Figure 5 presents a complementary analysis which canexplain the behavioral characteristics of the fuel consump-tion that is previously given. It gives, in particular, the thrustbehavior versus aircraft Mach number with bypass ratiovariation. The bypass ratio is varying between 0 and 6.The propulsive efficiency is confirmed. For any availableengine energy, dynamical thrust is optimized because of thebypass ratio optimization. This is interpreted by the rela-tionships in an action-reaction propulsion system. Thus, fora high bypass, the thrust is generally derived from the ductof the fan, rather than from combustion gases expandingin the engine nozzle. A high bypass ratio provides a lowerthrust specific fuel consumption (grams/sec fuel per unit ofthrust).

Low-bypass system is more fuel efficient and is muchquieter. Low bypass ratios do not tend to be favored for civil-ian aircraft because of the compromise between improvedfuel economy and the requirements of the performanceof the aircraft in terms of flight paths and its stability.The behavior of thrust is common and does not presentany sharp nor sudden fluctuations. Constraints on engineparameters can easily impact on the engine pressure andits temperature, and could be critical at some operatingpoints. This is why fuel consumption has a significant roleand is associated with engine performance. This associationhas the advantage of giving lower thrust values for extend-ing engine life and reducing loads. Engine performance ofmodern engines depends on requirements of accessories,engine size, inlet and duct design, air conditioning sys-tem compressor, hydraulic pumps power, alternators power,inlet and exhaust duct losses, ... Because of the technolog-ical development of these engines, thrust is reduced; butfuel consumption can increase. Optimization of the flightpath associated with new technological developments con-tribute significantly to the reduction of fuel consumption.Environmental impact and fuel consumption are reduced bythe use of optimization process for departures and arrivals.Operation management could be also improved by usingoptimized flight path. In addition, these flight path param-eters can be integrated in the flight management systemof the aircraft, and used in the autopilot system. This isa promising objective of this research. This bypass ratiostudy could contribute to provide numerical results and datafor the future generation engines which will have a higherbypass ratio. In association with passive control noise sys-tems, the primary advantage of this contribution should beengine efficiency analysis and weight savings. The secondadvantage, linked to the first, is the drag reduction. Thethird advantage, related to the increase of the bypass ratio,helps airline manufacturers to suggest the best arrangementof nacelle components to facilitate convenient removal andreplacement of engines without subjecting nacelle to highstresses. To conclude, optimization of flight path induces

fuel consumption savings, noise reduction and improvementof the engines-BPR knowledge.

4 Conclusion

Optimization model is expected to replace empirical mod-els for well-established applications such as predictingnoise contours around airports and fuel savings. Despitenumerical complexity, feasibility errors and time processingduration, this paper suggests a method which provides opti-mal trajectory and its parameters reducing noise and fuelconsumption. In this technical paper, we applied the bestnumerical method associated with an adequate algorithm wehave developed for solving aircraft trajectory optimizationproblem. It has taken into account jet noise source, fuel con-sumption, aircraft constraints and its operational extremelimits. Additionally, in spite of the fact that the maximuminput values of the aircraft speed, the throttle setting andaltitudes are kept free, the method gives reliable results.It is fast, errors are weak, and calculations are exact witha suitable convergence time. This new original approachcontributes to improve scientific knowledge in the field ofenvironmental impact reduction of aircraft.

The altitude decreases with three gradual slopes with twosegments. Aircraft alignment on the runway axis is per-formed with a slope of 3◦ during the last approach segment.Angle of descent is stable as recommended by ICAO andaircraft certification. The flight rate descent is about 1060ft/mn which is close to the one recommended by ICAO andpracticed by the airline companies.

This method characterizes the continuous descentapproach (CDA) which has the potential to reduce noiseemission and fuel consumption during the approach. Theobtained optimal trajectory could be accepted into the air-line community. The soft two-segment approach puts theaircraft in an appropriate envelope with margins for winduncertainties and errors. With autopilot or flight directorcoupling, this numerical method could be acceptable for usein regular air carrier service.

In addition, Mach number decreases during a short periodof the flight and remains stable during the whole approach.The aircraft finesse is bang-bang between its bounds. Ata constant speed, when the altitude decreases, the thrustincreases. It increases during the second flight period. If theguidance is satisfactory and the visual conditions are met,CDA is considered as a managed approach.

Evolution of the obtained noise levels under the flightpath, given by INM for standard trajectories in accordancewith ICAO specifications, is compared with the optimizedvalues behavior. By comparison, optimized noise levels arelower than those obtained by INM. In spite of the absenceof all noise models of different aircraft sources, calculated

Eur. Transp. Res. Rev. (2014) 6:71–84 81

values are close to experimental measurements. A decreasein noise levels can be confirmed when practicing the opti-mized flight path. A 4 dB reduction is obtained in favor ofthe optimization method.

In addition, we carried out experiments under the aircraftflight path to validate benefits of flight path optimization interm of noise reduction. Measurements of noise were per-formed under the flight path close to Saint-Exupery LyonInternational Airport during one year. We identified 15460turbojet aircraft executing approaches of the airport in thesame conditions: 84.5 % (+20 T) of the air traffic are tur-bojet aircraft, 15 % are propeller aircraft (3–9 T and +20T) and 0.5 % others (−3 T and 3–9 T). 360 aircraft areA300 or equivalent. Maximum noise levels vary from 91.9dB to 100.7 dB. These variations depend certainly on theatmospheric conditions, the aircraft loads, and the proce-dure variations initiated by the pilots. At the certificationpoint, average experimental values is 95.7 dB whereas themean levels obtained by INM and the optimization methodare respectively 93.7 dB and 89.3 dB. Optimal mean valueis largely below the measured and INM values. Thus, com-parisons show that optimal flight path favors noise levelreduction. This significant reduction could be regarded asover-estimated because the other noise models of fuselageare not introduced in the optimization model. Nevertheless,the optimization method is promising.

Engine performance is assessed by three calculationmethods. Using the optimization method, we have calcu-lated the ratio of aircraft thrust on the aircraft static thrustversus speed and by varying bypass ratio. Calculation ofthe fuel consumption is introduced as a second term ofthe cost function. In the open literature, specific fuel con-sumption have underestimated or overestimated the burnedfuel, or shown sudden changes. Optimization model pro-vides fuel consumption values between the two first modellimits. Optimal trajectory consumes less than the standardtrajectory. Comparison between models provided 10 % to20 % of fuel savings in favor of the optimized flight path.

The behavior of the thrust has no sudden fluctuations,confirming the propulsive efficiency. For available engineenergy, dynamical thrust is optimized because of the bypassratio improvement. This is interpreted by the relationships inan action-reaction propulsion system. Constraints on engineparameters easily impact the engine pressure, its temper-ature and could be critical at some operating points. Thisis why analysis of fuel consumption has a significant rolebecause it is associated with engine performance. Thisassociation has the advantage to lower thrust values forextending engine life and reducing loads.

To conclude, optimization of flight path associated withnew technological developments should contribute signifi-cantly to the reduction of noise and fuel consumption. Inparticular, association with passive control noise systems,

one of the advantages of this contribution should be en-gine efficiency analysis and weight savings. Environmen-tal impacts and fuel consumption are reduced by trajectoryoptimization of aircraft during arrivals. Optimized flightpath can be integrated in the flight management systemand can be used in the autopilot system. This is a promis-ing objective of this research. Further research is needed toinclude airframe noise sources, and air-brake systems.

Issues and alternatives Aviation industry is crucial to worldtrade. Its global economic impact is equivalent to more than7.5 per cent of world GDP. It directly employs more than5.5 million people world-wide. Besides economic impacts,aviation plays a pivotal role in connecting communities. Thelow cost flights have supported a strong demand growth.Neither reduction at source using quieter aircraft nor oper-ational restrictions (noise action plans, direct governmentregulation or voluntary airport initiatives) are delivering sat-isfactory mitigation. ICAO Balanced Approach to aircraftnoise control consists of identifying noise problems at anairport, then analyzing various measures. Long-term mea-sures must force the solutions at regional level: reductionof noise at source and certification; phase-out of non-certificated airplanes; noise charges; and land-use planningand management. Short-term measure must facilitate thesolutions at local level, like noise abatement proceduresand mitigation of aircraft operation. It is a major challengefor the future of air transport in the context of economicdevelopment linked to compliance with the conditions ofpeople living near airports. The land use planning elementof ICAO’s balanced approach is difficult to implement.

The use of advanced aircraft, optimized flight paths andimproved airspace management offer the most immediateways to mitigate aviation’s environmental impact. However,against growing demand for air transportation system theseefforts alone are unlikely to be sufficient for a significantimpact reduction in the long term. This paper gives a newchange to flight path approach which is characterized as anoise abatement technique. Pilots can descend at the ratebest suited to the achievement of continuous descent, withthe objective to join the glide path at the appropriate height.Rather than deploying flaps and descending through a givennumber of flight levels, the aircraft flight a continuoussteady descent at a fixed angle. It should be remembered thatP(recision)-RNAV in Europe, defined as operations whichsatisfy a required track keeping accuracy of 1 NM for atleast 95 % of the flight time, is not in contradiction with theoptimized flight path developed in this paper.

The main benefit of the optimized flight paths are thatthey avoid flaps deployment and thrust changes. The air-craft may also be higher at points along the approach path.Because of little few gear and flap changes, aircraft stayhigher along the descent segments contributing to noise

82 Eur. Transp. Res. Rev. (2014) 6:71–84

and fuel reduction. Because of environment benefits, theuse of the optimized flight path as a noise managementtool is advocated by regulators worldwide, from the ICAOto national governments. It also helps to reduce fuel con-sumption allowing an added value for airlines. Optimizedflight path is advised, promoted, and incorporated in a vol-untary code of practice compiled by airlines, air trafficcontrol, airport authorities and the transport departmentsof the European countries. This is a major action intend-ing to emphasize measures that can increase flight numberand avoiding air traffic saturation. Because of the growthin traffic, more flights can join the final approach withoutconflict and with less community annoyance. Thus, a com-promise should be found between environmental acceptabil-ity, the lower cost of design, development, production andexploitation, and increasing the operational capacity of theairspace.

Policy recommendations It is clear that neither quieter air-craft nor operational restrictions are satisfactory. Never-theless, optimized flight paths have potential environmen-tal and economical benefits compared to the present dayapproach procedures including:

• higher altitude during a large part of the approach,• lower power settings with clean aircraft configuration,• more flexibility in definition of approach path geome-

try, enabling the procedure designer to define approachpaths away from residential area

• additional advantage of the safety issue by reducingthird party risk.

There is sufficient evidence in this paper to support fur-ther assessment of optimized flight paths which provideflexibility where problems of flight concentration exist.Annoyance is strongly influenced by the number of aircraft.Noise policies and measures, taking into account the trafficincrease, have contributed to the communities living aroundairports. Limiting the number of the population impactedby aircraft noise can improve the quality of life for themand reduce the effect of concentrated flight paths. Noiseproblem has evolved over the last decades and become sen-sitive because it is now associated to air pollution problemsaffecting health population. Air traffic increase, new aircrafttechnology and airspace management associated to localaction by airport operators makes the environmental pol-icy reliable. Because of air traffic increase, the difficulty inmodifying flight paths location and protecting the popula-tion have all added to the complexity of this environmentalproblem. A new policy needs to consider these issues, anda perpetual revision should be a high priority. We recom-mend that the policy needs to include a general environ-mental duty in performing its operational actions. Airportauthorities and the policy need to have a sharp regard for

environmental factors alongside the safety and consumerobjectives. Failure to do this will leave airport authorities toexplain what is reasonable when taking decisions on matterssuch as airspace changes.

The preliminary question is what is the best way toachieve environment objectives associating technologicaldevelopment, the renewal of aircraft, implementation ofaircraft optimized procedures, airspace management, ... Inaddition to new environmental finding allowing to improveenvironmental impacts of aircraft, scientific communitiesneed to provide specific guidance relevant to air naviga-tion functions. Air traffic authorities also need to updatethe existing guidance, and European governments shouldextend a revision of the existing noise policy. Airport oper-ators, governmental environmental committees, airlines,air traffic managers and aircraft manufacturers should beactively engaged to initiate trials to assess the potential ben-efits of the combined possible solutions. Both governmentsand aircraft industries should address actions toward green-house gas emissions from aircraft and fuel saving ratherthan focusing on the only noise issues where emerging ideascould be developed.

Sustainability is a key issue for aviation which is unitedin its commitment to develop global solutions for the sus-tainable future of international civil aviation. With theflight paths to a sustainable future initiatives, ICAO hashighlighted the capability of national authorities, aircraftmanufacturers, airlines, air navigation service providers,and other stakeholders to work together to make this newflight paths possible. In addition, sustainability for air-ports means not only minimizing environmental impacts butalso communicating about the social and economic bene-fits of airports and aviation. Without taking into accountthe social requests, airports will not be able to meetfuture capacity demands. The following topics have to beconsidered:

- Safety and security (safety, sustainable developmentand control capacity, airport development, automaticassistance of control, ...);

- Management and user service;- European construction and preparation of the future

(Airspace, SESAR, support the implementation of thenew coming technologies of communication, naviga-tion, monitoring, ...);

- World politics of air traffic navigation.

They are declined in four major actions to support:

1. promoting research and development into new lownoise engine and airframe technologies;

2. implementing the regulatory framework agreed by theInternational Civil Aviation Organization (ICAO) usingthe balanced approach to noise management;

Eur. Transp. Res. Rev. (2014) 6:71–84 83

3. implementing EU Directives which require periodicnoise mapping;

4. the use of economic instruments which should have nolimitations for long-term benefits.

Annex

Perturbing the last two equations corresponding to thecomplementary conditions, by a positive parameter ε, weobtained:

This is the optimality conditions of the following problem:

(Pε)

⎧⎪⎪⎨

⎪⎪⎩

min∫ tf

0(�ε(z(t), u(t))+ φε(z(t), tf ))dt

z(t) = f (z(t), u(t)), t ∈ [0, tf ]�ε is the logarithmic barrier of (Pε):

�ε(z, u)+ φε(z, tf ) = �(z, u)+ φ(z, tf )

−ε∑

i

[log(Ci(z, u)− ai)

+ log(bi − Ci(z, u))]

−εD(z)

To solve (OC), we solved a sequence of problems (OCε)

by tending ε to zero. When ε decreases to 0, the solu-tion of optimal conditions (OCε) is a solution of (OC).Discretization of the continuous optimal conditions pro-vided a set of non-linear equations, which is solved forthe discretized control, state and costate vectors using theNewton method [37]. Euler schema provided for (OCε) thefollowing system:

We obtained a set of equations which are solved under theboundary constraints corresponding to the multiplicators:

(Nε)

⎧⎨

⎩

Fε(X) = 0λk ≥ 0μk ≤ 0

Fε is the set of optimal conditions, and X = (zk, uk,

pk, λk, μk) the variable vector. (OCP) is then successfullysolved for decreasing ε with a non-growth of the cost.

Open Access This article is distributed under the terms of theCreative Commons Attribution License which permits any use, dis-tribution and reproduction in any medium, provided the originalauthor(s) and source are credited.

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