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arXiv · 2412.13929

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

Ismaïla Balogoun·Jean Auriol·Islam Boussaada·Guilherme Mazanti
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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.

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Ismaïla Balogoun, Jean Auriol, Islam Boussaada, Guilherme Mazanti. 2024-12-18. A novel necessary and sufficient condition for the stability of $2\times 2$ first-order linear hyperbolic systems. https://doi.org/10.1016/j.sysconle.2025.106066

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