Deterministic optimal control on Riemannian manifolds under probability knowledge of the initial condition

Frédéric Jean (1) , Othmane Jerhaoui (1) , Hasnaa Zidani (2)

1 OC - Optimisation et commande
2 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie

In this article, we study an optimal control problem on a compact Riemannian manifold M with imperfect information on the initial state of the system. The lack of information is modelled by a Borel probability measure along which the initial state is distributed. The state space of this problem is the space of Borel probability measures over M. We define a notion of viscosity in this space by taking as test functions a subset of the set of functions that can be written as a difference of two semi-convex functions. With this choice of test functions, we extend the notion of viscosity solution to Hamilton-Jacobi-Bellman equations in Wasserstein space, we also establish that the value function of the control problem with imperfect information is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the space of Borel probability measures.

Mots clés: Hamilton-Jacobi equations in the Wasserstein space; viscosity solutions; differentiability in the Wasserstein space; optimal control theory

Domaines: Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

Frédéric Jean, Othmane Jerhaoui, Hasnaa Zidani. Deterministic optimal control on Riemannian manifolds under probability knowledge of the initial condition. 2022. ⟨hal-03564787v1⟩