Optimal Control for a Class of Infinite Dimensional Systems Involving an $L^\infty$-term in the Cost Functional

TL;DR

This paper studies an optimal control problem involving an $L^ Infty$-term in the cost functional, where the timing of the maximum state value is also optimized, and develops conditions and numerical methods for solution.

Contribution

It introduces a novel approach to handle the $L^ Infty$-term and free timing parameter in infinite-dimensional systems, deriving optimality conditions and implementing a Newton method.

Findings

Numerical simulations demonstrate the method's effectiveness.

The approach reveals the influence of the timing parameter on control strategies.

First and second-order optimality conditions are successfully derived.

Abstract

An optimal control problem with a time-parameter is considered. The functional to be optimized includes the maximum over time-horizon reached by a function of the state variable, and so an $L^\infty$-term. In addition to the classical control function, the time at which this maximum is reached is considered as a free parameter. The problem couples the behavior of the state and the control, with this time-parameter. A change of variable is introduced to derive first and second-order optimality conditions. This allows the implementation of a Newton method. Numerical simulations are developed, for selected ordinary differential equations and a partial differential equation, which illustrate the influence of the additional parameter and the original motivation.