Lyapunov stability of a singularly perturbed system of two conservation laws

Abstract

This paper is concerned with a class of singularly perturbed systems of two conservation laws. A small perturbation parameter is introduced in the dynamics and the boundary conditions. By setting the perturbation parameter to zero, the singularly perturbed system of conservation laws can be treated as two subsystems of one conservation law: the reduced system and the boundary-layer system. The asymptotic stability of the complete system is investigated via Lyapunov techniques. A Lyapunov function for the singularly perturbed system is obtained as a weighted sum of two Lyapunov functions of the subsystems.

Tikhonov theorem for linear hyperbolic systems

“…As this example is a linear singularly perturbed system of two conservation laws, it is not exponentially stable neither in L 2 -norm nor in H 2 -norm, although the reduced and boundary-layer systems are both exponentially stable. See also Tang et al (2013) where under the exponential stability of the both subsystems, an additional condition is introduced for the exponential stability of the linear singularly perturbed system.…”

A new H²-norm Lyapunov function for the stability of a singularly perturbed system of two conservation laws

“…In our previous work [9], a class of singularly perturbed system of two conservation laws with a small perturbation parameter introduced to both the dynamics and the boundary conditions has been studied. As soon as the two subsystems, the reduced and boundary-layer systems, are stable, the stability of this system can be obtained by a Lyapunov function in L 2 -norm.…”