Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

TL;DR

The paper addresses the stability analysis of a class of $2\times 2$ linear first-order hyperbolic PDEs with in-domain coupling by transforming the PDEs into an integral difference equation (IDE) via a backstepping transform. A St\'epan–Hassard–style spectral counting method is adapted to count unstable roots of the IDE through a characteristic function $\Delta(s)$, yielding a necessary and sufficient stability criterion. Specifically, stability is ensured when $|q\rho|<1$, $\Delta$ has no imaginary-axis zeros, and the counted number of unstable roots $\Gamma$ equals zero; these conditions transfer to the original PDE, providing an implementable stability test. Numerical validation and truncation analyses of the kernel function $N$ illustrate the practicality of the approach and its relation to prior sufficient conditions. The results offer a rigorous, verifiable framework for stability assessment and open avenues for extending the method to larger systems and multi-delay integral equations.

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 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.

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

TL;DR

The paper addresses the stability analysis of a class of linear first-order hyperbolic PDEs with in-domain coupling by transforming the PDEs into an integral difference equation (IDE) via a backstepping transform. A St\'epan–Hassard–style spectral counting method is adapted to count unstable roots of the IDE through a characteristic function , yielding a necessary and sufficient stability criterion. Specifically, stability is ensured when , has no imaginary-axis zeros, and the counted number of unstable roots equals zero; these conditions transfer to the original PDE, providing an implementable stability test. Numerical validation and truncation analyses of the kernel function illustrate the practicality of the approach and its relation to prior sufficient conditions. The results offer a rigorous, verifiable framework for stability assessment and open avenues for extending the method to larger systems and multi-delay integral equations.

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 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.

Paper Structure

This paper contains 15 sections, 10 theorems, 65 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem description
  3. System under consideration
  4. Objective and methodology
  5. Equivalent integral difference equation
  6. Main results
  7. Stability conditions for difference equation with distributed delay
  8. Application to the stability analysis of the system \ref{['eq:hyperbolic_couple']}
  9. Comparison with other criteria and numerical validation
  10. Numerical explorations of truncations of the function $N$
  11. Investigation of imaginary roots
  12. The case $p=0$
  13. The case $p=1$
  14. Stability of the truncated system
  15. Conclusion

Key Result

Theorem 1

There exist two positive constants $\kappa_0$ and $\kappa_1$ such that, for every $t>\tau$, Moreover, the exponential stability of $z_{[t]}$ in the sense of the $L^2(-\tau, 0)$ norm is equivalent to the exponential stability of $(\alpha,\beta)$ (or equivalently to $(u,v)$) in the sense of the $L^2$ norm.

Figures (8)

  • Figure 1: Roots of $\Delta$ with $N(\nu)=-\frac{3\pi^2}{16-4\pi}\nu + \frac{\pi^2+2\pi}{16 - 4\pi}$.
  • Figure 2: Graph of the function $M$ when $\left(\sigma^+, \sigma^-, \frac{1}{\lambda}, \frac{1}{\mu}, \rho, q\right)=(2.3,-3.5,0.8,1.1,0.5,-0.7)$.
  • Figure 3: $L^2$ norm of the solution $(u,v)$ of \ref{['eq:hyperbolic_couple']} with parameters $\left(\sigma^+, \sigma^-, \frac{1}{\lambda}, \frac{1}{\mu}, \rho, q\right)=(2.3, -3.5, 0.8, 1.1, 0.5, -0.7)$.
  • Figure 4: $N$, $N_0$, $N_1$, $N_2$, and $N_3$ with $\Bigl(\sigma^+, \sigma^-, \frac{1}{\lambda}, \frac{1}{\mu}, \rho, q\Bigr)=(2.3,-3.5,0.8,1.1,0.5,-0.7)$.
  • Figure 5: Solution of \ref{['distributed delay']} with $N$, $N_0$, $N_1$, $N_2$, and $N_3$ with $\Bigl(\sigma^+, \sigma^-, \frac{1}{\lambda}, \frac{1}{\mu}, \rho, q\Bigr)=(2.3, -3.5, 0.8, 1.1, 0.5, -0.7)$.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1: hale1993introduction, henry1974linear
  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 2
  • proof
  • Corollary 1
  • Remark 1
  • Remark 2
  • ...and 5 more