Linear-quadratic optimal control for infinite-dimensional input-state-output systems

TL;DR

This paper develops a framework for linear-quadratic optimal control of infinite-dimensional systems, including ill-posed PDEs, with applications to energy-efficient control of port-Hamiltonian systems and boundary-controlled PDEs.

Contribution

It introduces a novel approach for optimal control of a broad class of infinite-dimensional systems, including non-well-posed PDEs, using the system node formulation.

Findings

Established existence of solutions and optimality conditions for the control problem.

Applied the theory to energy-optimal control of port-Hamiltonian systems.

Illustrated the approach with examples involving heat and wave equations.

Abstract

We examine the minimization of a quadratic cost functional composed of the output and the final state of abstract infinite-dimensional evolution equations in view of existence of solutions and optimality conditions. While the initial value is prescribed, we are minimizing over all inputs within a specified convex subset of square integrable controls with values in a Hilbert space. The considered class of infinite-dimensional systems is based on the system node formulation. Thus, our developed approach includes optimal control of a wide variety of linear partial differential equations with boundary control and observation that are not well-posed in the sense that the output continuously depends on the input and the initial value. We provide an application of particular optimal control problems arising in energy-optimal control of port-Hamiltonian systems. Last, we illustrate the our abstract theory by two examples including a non-well-posed heat equation with Dirichlet boundary control and a wave equation on an L-shaped domain with boundary control of the stress in normal direction.