A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems

TL;DR

This paper derives a precise stability criterion for 2x2 first-order linear hyperbolic PDEs using a backstepping approach and a novel root-counting theorem, validated through simulations.

Contribution

It introduces a necessary and sufficient stability condition for these PDEs by linking them to an integral difference equation and developing a new root-counting theorem.

Findings

Established a necessary and sufficient stability condition.

Reformulated PDE stability as an integral difference equation problem.

Validated the theoretical criterion with simulations.

Abstract

In this paper, we establish a necessary and sufficient stability condition for a class of two coupled first-order linear hyperbolic partial differential equations. Through a backstepping transform, the problem is reformulated as a stability problem for an integral difference equation, that is, a difference equation with distributed delay. Building upon a St\'ep\'an--Hassard argument variation theorem originally designed for time-delay systems of retarded type, we then introduce a theorem that counts the number of unstable roots of our integral difference equation. This leads to the expected necessary and sufficient stability criterion for the system of first-order linear hyperbolic partial differential equations. Finally, we validate our theoretical findings through simulations.