Exponential Stability for Linear Evolutionary Equations

TL;DR

This paper presents a new approach to establishing exponential stability for linear evolutionary equations by deriving conditions based on the material law operator, with applications to various types of differential equations.

Contribution

It introduces sufficient conditions for exponential stability in evolutionary equations using an analytic operator-valued function framework, expanding stability analysis methods.

Findings

Derived explicit decay rate estimates.

Provided conditions applicable to differential-algebraic and delay equations.

Illustrated results with three diverse examples.

Abstract

We give an approach to exponential stability within the framework of evolutionary equations due to [R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci., 32(14):1768-1803,2009]. We derive sufficient conditions for exponential stability in terms of the material law operator, which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.