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

TL;DR

This work stabilizes a chain of three hyperbolic PDEs with the actuator located between the first and second subsystems by mapping the original system to a pure transport target via an invertible backstepping transformation. The stabilization problem is converted to an Integral Difference Equation (IDE) with time delays, which is then addressed using a dynamic, autoregressive feedback whose gains are obtained from Fredholm-type integral equations. Exponential stability is achieved under structural and spectral controllability assumptions, with a rigorous invertibility argument for the associated integral operator. The framework highlights a systematic path to extend stabilization to chains of arbitrary length and multiple actuators, enabling robust control of complex networked PDE systems.

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.

Figures (1)

• Figure 1: Schematic representation of system \ref{['system_edp_origin']}

Theorems & Definitions (10)

• Definition 2.1
• Remark 3.1
• Definition 3.2
• Remark 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 4.5
• proof
• Theorem 4.6
• Proposition B.1