Stabilization of a chain of 3 hyperbolic PDEs with 2 inputs in arbitrary position

TL;DR

This paper develops a unified backstepping-based method to stabilize a chain of three coupled hyperbolic PDEs with two inputs at arbitrary positions, revealing structural patterns and stability conditions.

Contribution

It introduces a novel unified framework using IDE reformulation for stabilizing multi-input hyperbolic PDE chains at arbitrary nodes, extending previous endpoint-focused results.

Findings

Most configurations require spectral controllability for stabilization.

One configuration can be stabilized without spectral conditions.

An explicit example shows spectral controllability can fail in some cases.

Abstract

This paper addresses the stabilization of a chain of three coupled hyperbolic partial differential equations actuated by two control inputs applied at arbitrary nodes of the network. With the exception of configurations where one input is located at an endpoint, cases already well studied in the literature, all admissible two-inputs configurations are treated in this paper within a unified framework. The proposed approach relies on a backstepping transformation combined with a reformulation of the closed-loop dynamics as an Integral Difference Equation (IDE). This IDE representation reveals a common structural pattern across configurations and clarifies the role played by delayed dynamics in the stability analysis. Within this formulation, the stabilization problem can be handled using existing IDE control techniques. For most configurations, the stabilization of the PDE system requires an approximate spectral controllability assumption. Remarkably, one specific configuration can be stabilized without imposing any additional spectral condition. In contrast, we also provide an explicit example of a configuration for which the required spectral controllability property fails to hold.