Value Function and Optimal Trajectories for Regional Control Problems via Dynamic Programming and Pontryagin Maximum Principles

Guy Barles (1, 2) , Ariela Briani (1, 2) , Emmanuel Trélat (3, 4)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson
3 CaGE - Control And GEometry
4 LJLL - Laboratoire Jacques-Louis Lions

In this paper we study the optimal trajectories for regional, deterministic optimal control problems, i.e., problems where the dynamics and the cost functional can be completely different in two regions of the state space and therefore present discontinuities at their interface. Our first aim is to prove the classical link between the study of optimality conditions and the Bell-man approach in this framework, and then exploit it to recover the value function under the assumption that optimal trajectories have only a countable number of switchings between the different regions. From an application point of view this is a very reasonable assumption because in practice one never implements chattering trajectories (i.e., enjoying the Zeno phenomenon). As a consequence of our description we obtain a new regularity result for the value function in the framework of hybrid optimal control problems.

Mots clés: Optimal control; Discontinuous dynamics; Bellman equation; Pontryagin maximum principle

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value Function and Optimal Trajectories for Regional Control Problems via Dynamic Programming and Pontryagin Maximum Principles. 2016. ⟨hal-01313559v1⟩