Upper and lower bounds of the value function for optimal control in the Wasserstein space

This paper explores the application of nonsmooth analysis in the Wasserstein space to finite-horizon optimal control problems for nonlocal continuity equations. We characterize the value function as a strict viscosity solution of the corresponding Bellman equation using the notions of $\varepsilon$-subdifferentials and $\varepsilon$-superdifferentials. The main paper's result is the fact that continuous subsolutions and supersolutions of this Bellman equation yield lower and upper bounds for the value function. These estimates rely on proximal calculus in the space of probability measures and the Moreau-Yosida regularization. Furthermore, the upper estimates provide a family of approximately optimal feedback strategies that realize the concept of proximal aiming.