DETERMINISTIC OPTIMAL CONTROL ON RIEMANNIAN MANIFOLDS UNDER PROBABILITY KNOWLEDGE OF THE INITIAL CONDITION

In this article, we study a Mayer optimal control problem on the space of Borel probability measures over a compact Riemannian manifold M. This is motivated by certain situations where a central planner of a deterministic controlled system has only imperfect information on the initial state of the system. The lack of information here is very specific. It is described by a Borel probability measure along which the initial state is distributed. We define a new notion of viscosity in this space by taking test functions that are directionally differentiable and can be written as a difference of two semiconvex functions. With this choice of test functions, we extend the notion of viscosity to Hamilton-Jacobi-Bellman equations in Wasserstein spaces and we establish that the value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the Wasserstein space over M.