Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach

The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) equation and Pontryagin's Minimum Principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization

Using switching detection and variational equations for the shooting method

International audienceWe study in this paper the resolution by single shooting of an optimal control problem with a bang-bang control involving a large number of commutations. We focus on the handling of these commutations regarding the precise computation of the shooting function and its Jacobian. We first observe the impact of a switching detection algorithm on the shooting method results. Then, we study the computation of the Jacobian of the shooting function, by comparing classical finite differences to a formulation using the variational equations. We consider as an application a low thrust orbital transfer with payload maximization. This kind of problem presents a discontinuous optimal control, and involves up to 1800 commutations for the lowest thrust. Copyright c 2000 John Wiley & Sons, Ltd

SHOOT2.0: An indirect grid shooting package for optimal control problems, with switching handling and embedded continuation

The SHOOT2.0 package implements an indirect shooting method for optimal control problems. It is specifically designed to handle control discontinuities, with an automatic switching detection that requires no assumptions concerning the number of switchings. Special care is also devoted to the computation of the Jacobian matrix of the shooting function, using the variational system instead of classical finite differences. The package also features an embedded continuation method and an automatic (parallel) grid shooting in order to reduce the dependency to the initialization

Shoot-1.1 Package - User Guide

This package implements a shooting method for solving boundary value problems, for instance resulting of the application of Pontryagin's Minimum Principle to an optimal control problem. The software is mostly Fortran90, with some third party Fortran77 codes for the numerical integration and non-linear equations system. Its features include the handling of right hand side discontinuities (such as caused by a bang-bang control) for the integration of the trajectory and the computation of Jacobians for the shooting method. The particular case of singular arcs for optimal control problems is also addressed

Singular arcs in the generalized Goddard's Problem

We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this report, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control

Optimal Design for Purcell Three-link Swimmer

International audienceIn this paper we address the question of the optimal design for the Purcell 3-link swimmer. More precisely we investigate the best link length ratio which maximizes its displacement. The dynamics of the swimmer is expressed as an ODE, using the Resistive Force Theory. Among a set of optimal strategies of deformation (strokes), we provide an asymptotic estimate of the displacement for small deformations, from which we derive the optimal link ratio. Numerical simulations are in good agreement with this theoretical estimate, and also cover larger amplitudes of deformation. Compared with the classical design of the Purcell swimmer, we observe a gain in displacement of roughly 60%

Bocop - A collection of examples

In this document we present a collection of classical optimal control problems which have been implemented and solved with Bocop. We recall the main features of the problems and of their solutions, and describe the numerical results obtained

Low thrust minimum-fuel orbital transfer: a homotopic approach

International audienceWe describe in this paper the study of an earth orbital transfer with a low thrust (typically electro-ionic) propulsion system. The objective is the maximization of the final mass, which leads to a discontinuous control with a huge number of thrust arcs. The resolution method is based on single shooting, combined to a homotopic approach in order to cope with the problem of the initial guess, which is actually critical for non-trivial problems. An important aspect of this choice is that we make no assumptions on the control structure, and in particular do not set the number of thrust arcs. This strategy allowed us to solve our problem (a transfer from Low Earth Orbit to Geosynchronous Equatorial Orbit, for a spacecraft with mass of 1500 kgs, either with or without a rendezvous) for thrusts as low as 0.1N, which corresponds to a one-year transfer involving several hundreds of revolutions and thrust arcs. The numerical results obtained also revealed strong regularity in the optimal control structure, as well as some practically interesting empiric laws concerning the dependency of the final mass with respect to the transfer time and maximal thrust

Résolution numérique de problèmes de contrôle optimal par une méthode homotopique simpliciale.

On s'intéresse ici à la résolution numérique de problèmes de contrôle optimal peu réguliers. On utilise à la base les méthodes dites indirectes, à la fois précises et rapides, mais en pratique parfois très sensibles à l'initialisation. Cette difficulté nous amène à utiliser une démarche homotopique, dans laquelle on part d'un problème apparenté plus facile à résoudre. Le "suivi de chemin" de l'homotopie connectant les deux problèmes, est ici réalisé par un algorithme de type simplicial. On s'intéresse en premier lieu à un problème de transfert orbital avec maximisation de la masse utile, puis à deux problèmes présentant des arcs singuliers. Les perspectives futures liées à ces travaux comprennent en particulier l'étude de problèmes à contraintes d'état, également délicats à résoudre par les méthodes indirectes. Par ailleurs, on souhaite comparer cette approche avec les méthodes directes, qui impliquent la discrétisation totale ou partielle du problème. ABSTRACT : This study deals with the numerical resolution of optimal control problems with a low regularity. We primarily use indirect methods, which are both fast and accurate, but suffer from a great sensitiveness to the initialization. This difficulty leads us to introduce a continuation approach, in which we start from a related, but easier to solve problem. The "path following" between the two problems is here implemented with a simplicial method. We first study an orbital transfer problem with payload maximization, then two singular arcs problems. The future perspectives related to this work include in particular the study of state constraints problems, which are difficult to solve with indirect methods. Also, we would like to compare this approach with direct methods, which imply total or partial discretization of the problem