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

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

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 dif...

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Veröffentlicht in:arXiv.org 2024-12
Hauptverfasser: Balogoun, Ismaïla, Mazanti, Guilherme, Auriol, Jean, Islam Boussaada
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Difference equations
Hyperbolic systems
Partial differential equations
Stability criteria
Theorems
Time delay systems
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Mazanti, Guilherme
Auriol, Jean
Islam Boussaada
description 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 Stépán--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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subjects Difference equations
Hyperbolic systems
Partial differential equations
Stability criteria
Theorems
Time delay systems
title A novel necessary and sufficient condition for the stability of \(2\times 2\) first-order linear hyperbolic systems
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