A Minimality Property of the Value Function in Optimal Control on Spaces of Probability Measures

An optimal control problem with (possibly) unbounded terminal cost is considered in P2(Rd), the space of Borel probability measures with finite second moment. We consider a suitable weak topology rendering P2(Rd) 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 (HJB) 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.