Exponential Stability of the Gurtin-Pipkin Equation with Memory by Spectral Analysis

In this article, we investigate the exponential stability of a special Gurtin-Pipkin system with memory. Firstly, the time-variant Gurtin-Pipkin equation is converted into a time-invariant system by some techniques, which allows us to able to analyze the spectral of system. Then, by spectral analysis and asymptotic analysis method, we calculate all the spectrum of the Gurtin-Pipkin system which consists of point spectrum and continuous spectrum. Moreover, it is proved that a sequence of generalized eigenfunctions of the system exists, which constitutes a Riesz basis for the state space and thus induces the growth bound of the semigroup. Finally, the exponential stability of the system is concluded.

I. Introduction

In the real world, the dynamic behaviors of systems in many branches of science and industry are depicted by various integro-differential equations because of memory behavior, such as [2], [8], [9]. Over the past several decades, a variety of researchers have paid attention to integro-differential equations with viscoelasticity or thermoviscoelasticity, see for instance [1], [6], [7], [10]. In [13], [14], a special basis property is proved: a sequence of generalized eigenfunctions of the system exists, which constitutes a Riesz basis for the state space. Other different studies for materials with viscoelasticity or thermoviscoelasticity can also be found in [3], [12].