Stability of switched linear hyperbolic systems by Lyapunov techniques (full version)

TL;DR

This paper investigates the exponential stability of switched linear hyperbolic PDE systems using Lyapunov methods, providing sufficient conditions that account for switching signals, dwell times, and characteristic velocities, supported by numerical simulations.

Contribution

It introduces new Lyapunov-based stability conditions for switched hyperbolic PDEs, considering various switching scenarios and characteristic velocities, extending existing stability analysis methods.

Findings

Derived sufficient stability conditions for switched hyperbolic systems.

Analyzed effects of dwell time and characteristic velocities on stability.

Validated results through numerical simulations.

Abstract

Switched linear hyperbolic partial differential equations are considered in this paper. They model infinite dimensional systems of conservation laws and balance laws, which are potentially affected by a distributed source or sink term. The dynamics and the boundary conditions are subject to abrupt changes given by a switching signal, modeled as a piecewise constant function and possibly a dwell time. By means of Lyapunov techniques some sufficient conditions are obtained for the exponential stability of the switching system, uniformly for all switching signals. Different cases are considered with or without a dwell time assumption on the switching signals, and on the number of positive characteristic velocities (which may also depend on the switching signal). Some numerical simulations are also given to illustrate some main results, and to motivate this study.