Work

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

Year: 2026

Type: article

Abstract: We consider$H_{\infty }$-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$H_{\infty }$norm are not conservative.

Source: IEEE Transactions on Automatic Control

DOI: 10.1109/tac.2026.3666695

Authors: Sachin Shivakumar, Amritam Das, Matthew Peet

Institutions: Los Alamos National Laboratory, Eindhoven University of Technology, Arizona State University

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation, Laboratory Directed Research and Development, Los Alamos National Laboratory

Awards: CMMI-EPSRC: A Unified Polynomial Optimization Framework for the Analysis and Control of Nonlinear Partial Differential Equations, War on Boundary Conditions - A Control-Oriented Framework for Partial Differential Equations, CPS: Small: A Convex Framework for Control of Interconnected Systems over Delayed Networks, Collaborative Research: Robust Control of Large-Scale Networks with Variable Delays using Partial Integral Equations