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 \(M\).