Stabilization of a Chain of Three Hyperbolic PDEs using a Time-Delay Representation

TL;DR

This paper presents a systematic method for stabilizing a chain of three hyperbolic PDEs by transforming the system into a transport form and designing a dynamic controller based on an associated IDE.

Contribution

It introduces a novel stabilization approach using backstepping and IDE reduction for a chain of hyperbolic PDEs, with gains computed via Fredholm integral equations.

Findings

Achieves exponential stabilization of the PDE chain.

Provides a systematic framework for controller design.

Transforms PDE stabilization into solving Fredholm integral equations.

Abstract

This paper addresses the stabilization of a chain system consisting of three hyperbolic Partial Differential Equations (PDEs). The system is reformulated into a pure transport system of equations via an invertible backstepping transformation. Using the method of characteristics and exploiting the inherent cascade structure of the chain, the stabilization problem is reduced to that of an associated Integral Difference Equation (IDE). A dynamic controller is designed for the IDE, whose gains are computed by solving a system of Fredholm-type integral equations. This approach provides a systematic framework for achieving exponential stabilization of the chain of hyperbolic PDEs.