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Méthodes géométriques et numériques en contrôle optimal et problèmes de Zermelo sur les surfaces et révolution : applications

Boris Wembe 1 
1 IRIT-APO - Algorithmes Parallèles et Optimisation
IRIT - Institut de recherche en informatique de Toulouse
Abstract : This work studies Zermelo problems on revolutions surfaces from the point of view of optimal control in the Hamiltonian framework by combining so-called geometrical and numerical methods. It is motivated by several case studies, in particular the historical example of Carathéodory-Zermelo which is one of the founding problems of the calculus of variations and an interesting normal form for the microlocal analysis of the global problem, as well as the so-called "vortex problem" which is a recent application from hydrodynamics and which describes the evolution of a passive particle around a vortex point. The considered Zermelo problem is determined by a triplet (M, g, F0) where M is a 2D manifold with normal coordinates q = (r,thêta), g is a Riemannian metric on M and F0 is a vector field defining the current (or the wind). From the optimal control view point, this problem corresponds to a minimum time transfer problem between two points q0 and q1 for an affine control system of the form : q(t) = F0(q) + u1F1(q) + u2F2(q), ‖u‖ <= 1 where q(t) ∈ M , u = (u1, u2) is the control and where the fields F1, F2 form an orthonormal reference frame associated to the metric g. This kind of problem already appears in Riemannian geometry with F0(q) ≡ 0 and in Finslerian geometry in the so-called weak case where the norm of the current (associated to the metric g) is < 1. The main goal of this study is to construct an optimal synthesis in a suitable neighborhood R which is a rectangle containing the initial point q0. This objective is summarized in the study of the regularity and the description of the (sub)-level lines, corresponding to the so-called Zermelian balls, of the minimal time value function Vq0 (q1) := inf {tf | q(tf,q0) = q1, where (q,u) is solution of the problem}. The main difficulty of this analysis lies in the existence of abnormal directions in the strong current case (i.e. the case where the norm of the current is > 1), which are associated to new situations such as : the loss of local controllability in q0, the discontinuity of the minimal time value function, the deformation of small spheres and balls, the apparition of new branches in the cut locus (set of points where trajectories lose their optimality), etc. Our analysis is essentially based on two points of view. The first point of view is the Caratheodory viewpoint, equivalent to the Goh transform in control. It allows us to rewrite the problem as an affine and scalar control system in dimension 3 and to use geometric tools of this framework in order to compute the conjugate and cut loci in relation with the local and global optimality of the solutions. The second point of view is the mechanical system viewpoint which allows to write the problem using a generalized potential and to use the Clairaut relation to integrate the flow and to classify the trajectories according to a classification method called GMR (generalized Morse-Reeb) which generalizes the Morse-Reeb classification in dynamical system. In this context, we introduce the concepts of Reeb foliation, Reeb component, separatrix geodesic which separated the geodesic flow into different classes. A special attention is given to the study of the vortex problem which can be seen as a toy model of the Kepler problem, but with an orthoradial current. In this case, we provide a result for the existence of an optimal solution and we construct the synthesis of the problem.
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Submitted on : Tuesday, March 8, 2022 - 11:22:08 AM
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  • HAL Id : tel-03601394, version 2

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Boris Wembe. Méthodes géométriques et numériques en contrôle optimal et problèmes de Zermelo sur les surfaces et révolution : applications. Variables complexes [math.CV]. Université Paul Sabatier - Toulouse III, 2021. Français. ⟨NNT : 2021TOU30126⟩. ⟨tel-03601394v2⟩

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