Optimal Control of Moving Sets

TL;DR

This paper develops an optimal control framework for moving sets, aiming to minimize contaminated area over time with boundary control, providing existence, necessary conditions, and explicit solutions.

Contribution

It introduces a novel optimal control model for moving sets with boundary speed control, including existence proofs and Pontryagin maximum principle conditions.

Findings

Existence of optimal solutions within finite perimeter sets.

Necessary optimality conditions derived via Pontryagin maximum principle.

Explicit solutions illustrating the control of moving sets.

Abstract

Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets $t\mapsto \Omega(t)\subset\mathbb{R}^2$. Given an initial set $\Omega_0$, the goal is to minimize the area of the contaminated set $\Omega(t)$ over time, plus a cost related to the control effort. Here the control function is the inward normal speed along the boundary $\partial \Omega(t)$. We prove the existence of optimal solutions, within a class of sets with finite perimeter. Necessary conditions for optimality are then derived, in the form of a Pontryagin maximum principle. Additional optimality conditions show that the sets $\Omega(t)$ cannot have certain types of outward or inward corners. Finally, some explicit solutions are presented.