On the role of abnormal minimizers in sub-riemannian geometry

Bonnard, Bernard; Trélat, Emmanuel

doi:10.5802/afst.998

Cited by 22 publications

(28 citation statements)

References 28 publications

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“…This means that we have to ensure that the exponential mapping is proper, uniformly with respect to λ. The properness of the exponential mapping has been studied in [25], where it has been proved that, if the exponential mapping is not proper, then there must exist an abnormal minimizer (see also [168], and see [21, Lemma 2.16] for a more general statement). By contraposition, if one assumes the absence of minimizing abnormal extremals, then the required boundedness follows.…”

mentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

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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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mentioning

confidence: 99%

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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“…3.2.1 Geometric structure for the simplified cost and relation with Takagi's sinusoidal paddling (11) Note that if the cost is simplified to…”

Section: The Normal Casementioning

confidence: 99%

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

Bettiol¹,

Bonnard²,

Rouot³

2018

SIAM J. Control Optim.

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“…First of all using expression (11) it is easy to prove point 2 of Theorem 2.10. Indeed to study the accessibility set at time T from 0 we have to consider functions ξ such that:…”

Section: Denotes the Symmetric Bilinear Form Associated To The Quadramentioning

confidence: 99%

Asymptotics of accessibility sets along an abnormal trajectory

Trélat

2001

ESAIM: COCV

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Abstract. We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L ∞ -sector and the L 2 -sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.Mathematics Subject Classification. 93B03, 49K15.

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On the role of abnormal minimizers in sub-riemannian geometry

Cited by 22 publications

References 28 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Strokes at Low Reynolds Number: A Geometric and Numerical Study of Copepod and Purcell Swimmers

Asymptotics of accessibility sets along an abnormal trajectory

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