Contrôle Optimal

Objectives

We are interested in the numerical resolution of optimal control problems in the ordinary differential equations. An optimal control problem is an optimisation problem in infinite dimension whose unknown (the control) is a function of time. Such problems are coming from many applications: orbital transfer, medical imaging... The goal is to solve efficiently these problems via indirect shooting methods.

Description

- Mathematical formulation and examples of optimal control problems

- Necessary conditions of optimality

- Presentation of the numerical solver: the indirect shooting method

- Application to the orbital transfer problem minimising the energy

Complements will be given:

- in differential calculus in Banach spaces

- in the theory of ordinary differential equations

The practical work will be done in the Julia language (similar to Matlab but free) with a quick presentation.  The student:

- will implement Runge-Kutta schemes for numerical integration

- will implement an indirect shooting method

- will see the importance of the calculation of derivatives (by finite differences, variational equations, automatic differentiation) in order to improve the convergence of the shooting method

Bibliography

-    Dérivation, intégration, édition revue et augmentée, C.Wagschal, Hermann éditeurs, 2012

-    Analyse numérique et équations différentielles, J.-P. Demailly, Presses Universitaires de Grenoble, 1996

-    Contrôle optimal : théorie & applications, E. Trélat, Vuibert, Collection "Ma- thématiques Concrètes", 2005

Needed prerequisite

Optimisation en dimension finie

Intégration

Session 1 ou session unique - Contrôle des connaissances

Session 2 - Contrôle des connaissances