An algorithmic guide for finite-dimensional optimal control problems

Jean-Baptiste Caillau (1, 2, 3) , Roberto Ferretti (4) , Emmanuel Trélat (5, 6) , Hasnaa Zidani (7, 8)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 McTAO - Mathematics for Control, Transport and Applications
3 UniCA - Université Côte d'Azur
4 Dipartimento di Matematica e Fisica [Roma]
5 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
6 CaGE - Control And GEometry
7 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
8 NorMath - Fédération Normandie Mathématiques - Normandie Mathématiques

We survey the main numerical techniques for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider two classical examples, simple but significant enough to be enriched and generalized to other settings: Zermelo and Goddard problems. We provide sample of the codes used to solve them and make these codes available online. We discuss direct and indirect methods, Hamilton-Jacobi approach, ending with optimistic planning. The examples illustrate the pros and cons of each method, and we show how these approaches can be combined into powerful tools for the numerical solution of optimal control problems for ordinary differential equations.

Mots clés: Optimal control; Dynamical systems; Pontryagin maximum principle; Direct and indirect methods; Hamilton-Jacobi-Bellman equation; Optimistic planning algorithms

Domaines: Optimisation et contrôle [math.OC]

Jean-Baptiste Caillau, Roberto Ferretti, Emmanuel Trélat, Hasnaa Zidani. An algorithmic guide for finite-dimensional optimal control problems. Handbook of numerical analysis: Numerical control, Part B, 24, North-Holland; Elsevier, pp.559-626, 2023, Handbook of Numerical Analysis, ⟨10.1016/bs.hna.2022.11.006⟩. ⟨hal-03883822⟩