Viscosity Solutions for Infinite-Dimensional HJB Equations under State Constraints

We consider a finite-horizon optimal control problem in which the state variable is a probability density representing a population of agents, evolving according to a continuity equation driven by a controlled vector field belonging to a class of admissible controls. The novelty of the model is an integral state constraint, forcing the mass to remain confined within a bounded and sufficiently regular subset of the underlying space. A Bolza-type functional assigns a cost to each admissible state–control pair. We introduce the value function and prove its Lipschitz regularity. We then derive the associated infinite-dimensional Hamilton–Jacobi–Bellman (HJB) equation and an appropriate notion of viscosity solution, formulated as a subsolution in the interior of the admissible set and a supersolution up to its boundary. We show that the value function is a viscosity solution of the HJB equation. Finally, exploiting a geometric inward-pointing cone property, we prove a comparison theorem ensuring uniqueness - within the class of Lipschitz functions - of the solution to the Cauchy problem for the HJB equation.