Jean-Baptiste Caillau

Differential continuation for regular optimal control problems

Regular control problems in the sense of the Legendre condition are defined, and second-order necessary and sufficient optimality conditions in this class are reviewed. Adding a scalar homotopy parameter, differential pathfollowing is introduced. The previous sufficient conditions ensure the definiteness and regularity of the path. The role of AD for the implementation of this approach is discussed, and two examples excerpted from quantum and space mechanics are detailed.

Minimum time control of the restricted three-body problem

The minimum time control of the circular restricted three-body problem is considered. Controllability is proved on an adequate submanifold. Singularities of the extremal flow are studied by means of a stratification of the switching surface. Properties of homotopy maps in optimal control are framed in a simple case. The analysis is used to perform continuations on the two parameters of the problem: the ratio of masses and the magnitude of the control.

L 1 -minimization for mechanical systems

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the

Introduction to nonlinear optimal control

{\mathrm{L^1}}

Geodesic flow of the averaged controlled Kepler equation

-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [H. M. Robbins, AIAA J., 3 (1965), pp. 1094--1098; M. I. Zelikin and V. F. Borisov, Theory of Chattering Control, Birkhauser, Basel, 1994]; the case of the two-body potential is treated in detail. In

Averaging and optimal control of elliptic Keplerian orbits with low propulsion

{\mathrm{L^1}}

Sensitivity analysis for time optimal orbit transfer

-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [J. Noble and H. Schattler, J. Math. Anal. Appl., 269 (2002), pp. 98--128]. Two examples of numerical verification of these conditions are proposed on a problem coming from space mechanics.

On Some Riemannian Aspects of Two and Three-Body Controlled Problems*

The maximum principle is presented in the weak and general forms. The standard proofs are detailed, and the connection with the shooting method for numerical resolution is made. A brief introduction to the micro-local analysis of extremals is also provided. Regarding second-order conditions, small time-optimality is addressed by means of high order generalized variations. As for local optimality of extremals, the conjugate point theory is introduced both for regular problems and for minimum time singular single input affine control systems. The analysis is applied to the minimum time control of the Kepler equation, and the numerical simulations for the corresponding orbit transfer problems are given. In the case of state constrained optimal control problems, necessary conditions are stated for boundary arcs. The junction and reflection conditions are derived in the Riemannian case.

On Local Optima in Minimum Time Control of the Restricted Three-Body Problem

Abstract A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to S 2 is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity at the origin in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controlled Kepler system are finally obtained thanks to the computation of the cut locus of the restriction to the sphere.

Computation of conjugate times in smooth optimal control: the COTCOT algorithm

This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion. It is based on preliminary results of Geffroy [Generalisation des techniques de moyennation en controle optimal, application aux problemes de rendez-vous orbitaux a poussee faible, Ph.D. Thesis, Institut National Polytechnique de Toulouse, France, Octobre 1997] where the optimal trajectories are approximated using averaging techniques. The objective is to introduce the appropriate geometric framework and to complete the analysis of the averaged optimal trajectories for energy minimization, showing in particular the connection with Riemannian problems having integrable geodesics.