Genericity results for singular curves

Chitour, Yacine; Jean, Frédéric; Trélat, Emmanuel

doi:10.4310/jdg/1146680512

Cited by 118 publications

(112 citation statements)

References 29 publications

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“…We refer the reader to [19] for more details on this theory. It can also be shown that this kind of computation is valid in a "generic" situation (see [44,45,46,47]). …”

Section: Practical Use Of the Pontryagin Maximum Principlementioning

confidence: 95%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

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189

179

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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.

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“…We refer the reader to [19] for more details on this theory. It can also be shown that this kind of computation is valid in a "generic" situation (see [44,45,46,47]). …”

Section: Practical Use Of the Pontryagin Maximum Principlementioning

confidence: 95%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

Self Cite

189

179

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“…Let us moreover observe that, given a medium-fat distribution, it can be shown that for a generic smooth complete Riemannian metric on M the distribution does not admit nontrivial singular sub-Riemannian minimizing geodesics (see [15,16]). As a consequence, we have: Proposition 4.7.…”

Section: Medium-fat Distributionsmentioning

confidence: 99%

“…Denote by D m the space of rank m distributions on M endowed with the Whitney C ∞ topology. Chitour, Jean and Trélat proved that there exists an open dense subset O m of D m such that every element of O m does not admit nontrivial minimizing singular paths (see [15,16]). As a consequence, we have:…”

Section: Generic Sub-riemannian Structuresmentioning

confidence: 99%

Mass Transportation on Sub-Riemannian Manifolds

Figalli

Rifford

2010

Geom. Funct. Anal.

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We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Ampère equation.

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“…4. That is why the existence or non-existence of abnormal minimizers is only known for specific control problems, mainly control-linear and control-affine systems with control-quadratic cost functions or for time-optimal control problems [4,6,19,26,27,77].…”

Section: Introductionmentioning

confidence: 99%

Geometric Approach to Pontryagin’s Maximum Principle

Barbero-Lin͂án¹,

Muñoz-Lecanda²

2008

Acta Appl Math

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Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self-contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.

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Genericity results for singular curves

Cited by 118 publications

References 29 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Mass Transportation on Sub-Riemannian Manifolds

Geometric Approach to Pontryagin’s Maximum Principle

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