Principles of Optimal Control Theory

Гамкрелидзе, Р. В.

doi:10.1007/978-1-4684-7398-8

Cited by 225 publications

(164 citation statements)

References 0 publications

Supporting

Mentioning

161

Contrasting

Unclassified

Order By: Relevance

“…The p 0 still appears in the transversality condition (11). Note that, if M 1 = IR n (no condition on the final point), then this condition leads to p(t f ) = p 0 ∂g ∂x (t f , x(t f )), and then necessarily p 0 = 0 (otherwise the adjoint vector (p(t f ), p 0 ) would be zero and this would contradict the assertion of the Pontryagin Maximum Principle) and in that case we can normalize to p 0 = −1.…”

Section: Pontryagin Maximum Principlementioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

View full text Add to dashboard Cite

This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

show abstract

Section: Pontryagin Maximum Principlementioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

View full text Add to dashboard Cite

show abstract

“…When restricting the quadratic form (2.16) to the kernel of ( , F )|û (·) , determined by the equalities 17) we are able to subtract from (2.16) a vanishing value of…”

Section: Second Variation Along Normal and Abnormal Geodesicsmentioning

confidence: 99%

“…One can find in [17] examples of using this norm, e.g. a theorem on continuous, with respect to this norm, dependence of solutions of ODE on right-hand side.…”

Section: Proof Of the Theorem 43 On Compactness Of The Set Of Minimamentioning

confidence: 99%

Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Agrachev

Sarychev

1999

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. Analyse nonlinéaire 13, p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize these classes of distributions we determine the dimensions of the manifolds on which generic germs of distributions of given rank are respectively 2-generating or medium fat.Résumé. Onétudie des métriques sous-Riemanniennes (des Carnot-Carathéodory) définies par les distributions non involutives sur les variétés Riemanniennes analytiques réelles. Onétablie la connexion entre les propriétés de la régularité de ces métriques et l'absence des géodésiques anormales de longueur minimale. En utilisant les résultats desétudes précédentes sur les minimiseurs anormaux accomplies par les auteurs dans [Annales IHP. Analyse nonlinéaire 13, p. 635-690], on décrit dans cet article, pour certains types des germes, des distributions (appelées 2-générées et d'une croissance moyenne) telles que les métriques sous-Riemanniennes correspondantes sont sous-analytiques. Pour caractériser la classe de ces types des distributions, on détermine les dimensions des variétés sur lesquelles les germes génériques des distributions du rang donné sont 2-générées ou d'une croissance moyenne.

show abstract

“…We assume that the problem has an optimal solution with an optimal cost. In this paper, we have applied an indirect method [27][28][29][30], based on Pontryagin's principle [33][34][35][36] assessing the solutions. Inequality constraints are carried out by Pontryagin's maximum principle.…”

Section: γ (T) χ(T) X(t) Y(t) H(t)]mentioning

confidence: 99%

“…The new unconstrained (OCP ) is obtained by having the same system dimension. To solve (OCP ) problems, many methods exist in the open literature [31][32][33][34][35][36]. In this paper, we have used the optimal control technique developed in the references [16][17][18][19] which we have summarized.…”

Section: γ (T) χ(T) X(t) Y(t) H(t)]mentioning

confidence: 99%

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

Khardi¹

2013

Eur. Transp. Res. Rev.

View full text Add to dashboard Cite

Introduction Flight path optimization is designed for minimizing environmental impacts of aircraft around airports during approaches. The main objective of this paper is to develop a model of optimal flight paths taking into account jet noise, fuel consumption, constraints and extreme operational limits of the aircraft in approach. Materials and methods Optimal control problem combining aircraft flight paths, noise and fuel consumption is modeled and solved. Outputs, algorithm efficiency and their ability to be interfaced with the in-flight management system, respecting airspace system regulation constraints, are analyzed. Measurements of aircraft noise are carried out around Saint-Exupery Lyon International Airport for one year, and INM simulations are performed. Results and conclusion A two-segment approach is considered as an optimal trajectory. Aircraft alignment on the runway axis with a slope between 3 • and 4.5 • during the last approach segment is obtained and the descent rate is about 1060 ft/mn. This particularly characterizes the continuous descent approach having the potential to reduce noise emission 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 INM values given by the empirical trajectory. Optimal trajectory consumes less than the standard one; and it can be integrated in the aircraft FMS and the autopilot system. This is one of

show abstract

Principles of Optimal Control Theory

Cited by 225 publications

References 0 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

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

Contact Info

Product

Resources

About