LQ optimal control for infinite-dimensional passive systems

TL;DR

This paper investigates the Linear-Quadratic optimal control problem for a broad class of infinite-dimensional passive systems, providing new theoretical insights and explicit solutions, especially for energy-preserving cases and practical boundary control systems.

Contribution

It establishes conditions under which the finite cost condition holds, characterizes the optimal cost operator as a contraction, and derives explicit solutions for energy-preserving systems.

Findings

The finite cost condition is always satisfied under mild assumptions.

The optimal cost operator is a contraction, simplifying analysis.

Explicit solutions are derived for energy-preserving systems, including boundary control and port-Hamiltonian systems.

Abstract

We study the Linear-Quadratic optimal control problem for a general class of infinite-dimensional passive systems, allowing for unbounded input and output operators. We show that under mild assumptions, the finite cost condition is always satisfied. Moreover, we show that the optimal cost operator is a contraction. In the case where the system is energy preserving, the optimal cost operator is shown to be the identity, which allows to deduce easily the unique stabilizing optimal control. In this case, we derive an explicit solution to an adapted operator Riccati equation. We apply our results to boundary control systems, first-order port-Hamiltonian systems and an Euler-Bernoulli beam with shear force control.