Optimal control of the Poisson equation with transport regularization: Properties of optimal transport plans and transport map

TL;DR

This paper studies an optimal control problem involving the Poisson equation with a transport regularization term, analyzing the properties of optimal solutions and transport plans to inform control structure.

Contribution

It introduces a novel control framework combining Poisson equation constraints with transport regularization and derives optimality conditions and structural insights.

Findings

Existence of optimal solutions established

First-order optimality conditions derived

Structural properties of optimal controls and transport plans analyzed

Abstract

An optimal control problem in the space of Borel measures governed by the Poisson equation is investigated. The characteristic feature of the problem under consideration is the Tikhonov regularization term in form of the transportation distance of the control to a given prior. Existence of optimal solutions is shown and first-order necessary optimality conditions are derived. The latter are used to deduce structural a priori information about the optimal control and its support based on properties of the associated optimal transport plan.