Time-optimal problem in the space of probabilities measures

TL;DR

This paper investigates the time-optimal control problem for a continuity equation in probability measure space, establishing a dynamic programming principle and viscosity solution characterization, with convergence results for perturbed problems.

Contribution

It introduces a novel approach to analyze the value function in the space of probability measures, including viscosity solutions and convergence analysis.

Findings

The Kruzhkov transform of the value function is a unique discontinuous viscosity solution.

Established the dynamic programming principle for the problem.

Proved $\Gamma$-convergence of perturbed value functions to the unperturbed case.

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 is a unique discontinuous viscosity solution to the corresponding Dirichlet problem for the Hamilton-Jacobi equation. Finally, we establish the $\Gamma$-convergence of the value function in a perturbed problem to the value function in the unperturbed problem.