Optimal control of ODEs with state suprema

TL;DR

This paper addresses the optimal control of ODEs involving state supremum functionals, introducing a novel regularization approach to derive optimality conditions and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a new regularization method using LogIntExp for non-smooth state supremum problems, enabling derivation of optimality systems.

Findings

Existence of solutions proved for the control problem.

Regularization via LogIntExp allows derivation of optimality conditions.

Numerical experiments validate the theoretical approach.

Abstract

We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a novel LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.