Optimal Control of Continuity Equations

TL;DR

This paper studies an optimal control problem for the continuity equation, aiming to maximize mass in a target set, and establishes existence, necessary conditions, and approximation methods for optimal controls.

Contribution

It introduces a framework for controlling the continuity equation, proves existence of optimal controls, derives necessary conditions in special cases, and proposes approximation techniques.

Findings

Existence of optimal controls is proven.

Necessary optimality conditions are derived for smooth cases.

Perturbed problems can approximate solutions for the general case.

Abstract

An optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.