Optimal Boundary Control of 2x2 Linear Hyperbolic PDEs

TL;DR

This paper develops an optimal boundary control strategy for 2x2 linear hyperbolic PDEs, deriving necessary conditions and presenting a state-feedback controller based on Riccati equations, with numerical validation.

Contribution

It introduces a novel LQR boundary control method for hyperbolic PDEs with actuation at one boundary, including derivation of optimality conditions and controller design.

Findings

Controller effectively stabilizes the PDE system.

Numerical examples demonstrate the method's applicability.

The approach provides a systematic way to design boundary controllers.

Abstract

The present paper develops an optimal linear quadratic boundary controller for $2\times2$ linear hyperbolic partial differential equations (PDEs) with actuation on only one end of the domain. First-order necessary conditions for optimality is derived via weak variations and an optimal controller in state-feedback form is presented. The linear quadratic regulator (LQR) controller is calculated from differential algebraic Riccati equations. Numerical examples are performed to show the use of the proposed method.