A Minimality Property of the Value Function in Optimal Control over the Wasserstein Space

In this talk we study an optimal control problem with (possibly) unbounded terminal cost in the space of Borel probability measures with finite second moment. We consider the metric geometry associated with the Wasserstein distance, and a suitable weak topology rendering this space locally compact. In this setting, we show that the value function of a control problem is the minimal viscosity supersolution of an appropriate Hamilton-Jacobi-Bellman equation. Additionally, if the terminal cost is bounded and continuous, we show that the value function is the unique viscosity solution of the HJB equation.