Optimal Solutions for a Class of Set-Valued Evolution Problems

TL;DR

This paper characterizes optimal control strategies for the evolution of convex sets over time, showing that activating control along maximal curvature boundary parts is optimal, ensuring convexity throughout the process.

Contribution

It provides a novel optimality criterion for set evolution problems, linking boundary curvature to control activation in a convex set context.

Findings

Optimal control activates along maximal curvature boundary segments.

Convexity of the set is preserved over time.

Optimal strategies are characterized by boundary curvature analysis.

Abstract

The paper is concerned with a class of optimization problems for moving sets $t\mapsto\Omega(t)\subset\mathbb{R}^2$, motivated by the control of invasive biological populations. Assuming that the initial contaminated set $\Omega_0$ is convex, we prove that a strategy is optimal if an only if at each given time $t\in [0,T]$ the control is active along the portion of the boundary $\partial \Omega(t)$ where the curvature is maximal. In particular, this implies that $\Omega(t)$ is convex for all $t\geq 0$. The proof relies on the analysis of a one-step constrained optimization problem, obtained by a time discretization.