Dual Representations and $H_{\infty}$-Optimal Control of Partial Differential Equations

TL;DR

This paper develops a new approach for $ ext{H}_ ext{infty}$-optimal control of linear PDEs using PIE representations, enabling convex optimization methods and duality principles similar to ODE control.

Contribution

It introduces the concept of dual PIEs for PDEs, allowing the reformulation of control problems as convex optimization over positive PI operators.

Findings

Dual PIEs preserve stability and I/O properties of the original PDEs.

Convex optimization over PI operators yields less conservative $ ext{H}_ extinfty$ bounds.

Constructed feedback gains effectively stabilize PDE systems.

Abstract

We consider $\hinf$-optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs and indicate the resulting bounds on $\hinf$ norm are not conservative.