Time-optimal problem in the space of probabilities measures

Abstract

This paper focuses on the value function in the time-optimal problem for a continuity equation in the space of probability measures. We derive the dynamic programming principle for this problem. In particular, we prove that the Kruzhkov transform of the value function coincides with a unique discontinuous viscosity solution to the corresponding Dirichlet problem for the Hamilton-Jacobi equation. Finally, we establish the $Γ$-convergence of the value function in a perturbed problem to the value function in the unperturbed problem.