State-constrained optimal control of the continuity equation and infinite-dimensional viscosity solutions

TL;DR

This paper investigates a finite horizon optimal control problem for the continuity equation with state constraints, establishing the Lipschitz continuity of the value function and its characterization as a viscosity solution of the Hamilton-Jacobi-Bellman equation.

Contribution

It introduces a Hilbert space framework for the problem, proves the value function's regularity, and demonstrates the uniqueness of viscosity solutions under constraints.

Findings

Value function is Lipschitz continuous on the invariant set.

The value function satisfies the constrained Hamilton-Jacobi-Bellman equation in viscosity sense.

Comparison principle and uniqueness are established for the viscosity solutions.

Abstract

We study a finite horizon optimal control problem for the continuity equation under a weighted integral state constraint on the mass outside a fixed set. The model is cast in a Hilbert framework for densities. On a suitable invariant compact subset, we prove that the value function is Lipschitz continuous and satisfies, by dynamic programming, the associated infinite dimensional constrained Hamilton Jacobi Bellman equation in viscosity sense (subsolution in the interior, supersolution up to the boundary). We finally prove a comparison principle and uniqueness in the Lipschitz class.