Robust stabilization of $2 \times 2$ first-order hyperbolic PDEs with uncertain input delay

TL;DR

This paper develops a backstepping-based control method for stabilizing 2x2 hyperbolic PDE systems with uncertain boundary input delay, ensuring robustness through a novel transformation and sensitivity analysis.

Contribution

A new backstepping compensator design for hyperbolic PDEs with uncertain delays using coupled PDE transformations and robustness analysis.

Findings

The control is robust to small delay variations.

The method ensures well-posedness of the coupled PDE kernel equations.

Numerical examples demonstrate effective stabilization.

Abstract

A backstepping-based compensator design is developed for a system of $2\times2$ first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.