Optimal control of infinite-dimensional dissipative systems

TL;DR

This paper investigates the linear-quadratic optimal control problem for infinite-dimensional dissipative systems, establishing equivalence under certain conditions and connecting it to the operator Lur'e equation, with practical examples.

Contribution

It introduces a method to handle indefinite cost functionals in infinite-dimensional systems by relating them to nonnegative cost problems and explores the associated operator equations.

Findings

Equivalence between indefinite and nonnegative cost control problems.

Relationship established between value functions and the operator Lur'e equation.

Illustrative examples demonstrating the theoretical results.

Abstract

We study the linear-quadratic optimal control problem for infinite-dimensional dissipative systems with possibly indefinite cost functional. Under the assumption that a storage function exists, we show that this indefinite optimal control problem is equivalent to a linear-quadratic optimal control problem with a nonnegative cost functional. We establish the relationship between the corresponding value functions and present the associated operator Lur'e equation. Finally, we illustrate our results with several examples.