New Lyapunov functions for systems with source terms

TL;DR

This paper introduces a new class of Lyapunov functions with hyperbolic weights for stability analysis of hyperbolic systems with source terms, expanding the range of parameters for which exponential stability can be established.

Contribution

It extends classical exponential weights to hyperbolic functions, improving stability analysis for $2\times 2$ systems and boundary feedback control of hyperbolic PDEs.

Findings

New hyperbolic Lyapunov functions provide stability over larger parameter sets.

Large time-delays reduce the stabilizability region.

Hyperbolic weights are effective in boundary feedback stability analysis.

Abstract

Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of $2\times 2$ systems of balance laws. We present cases connected with the study of the limit of stabilizability where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights. Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set where the system can be stabilized becomes substantially smaller. We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations.