Pattern-preserving optimal control problems with increasing time horizon

TL;DR

This paper develops a framework ensuring that optimal control patterns are preserved as the time horizon extends indefinitely, with applications to switched systems and epidemic control.

Contribution

It introduces the concept of pattern-preserving families of optimal control problems and analyzes their behavior via $$-convergence as the horizon grows.

Findings

Established conditions for pattern preservation in increasing horizons.

Applied the framework to switched systems and epidemic control scenarios.

Provided examples demonstrating the theoretical results.

Abstract

We establish a general framework that guarantees the preservation of optimal control patterns as the time horizon $[0,T]$ increases and becomes unbounded. A concept of pattern-preserving family of optimal control problems is introduced and the goal is achieved by analyzing the $\Gamma$-convergence of the corresponding variational formulations as $T\to\infty$. Special attention is given to scenarios involving state constraints. To illustrate the results, examples and applications are provided, with particular focus on switched systems and epidemic control.