Stability of the heat and of the wave equations with boundary time varying delays Serge Nicaise Julie Valein† Emilia Fridman‡

Description

Niveau: Supérieur, Doctorat, Bac+8
Stability of the heat and of the wave equations with boundary time-varying delays Serge Nicaise?, Julie Valein†, Emilia Fridman‡ February 14, 2008 Abstract Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be time- varying with an a priori given upper bound on its derivative, which is less than 1. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given. Keywords heat equation, wave equation, time-varying delay, stability, Lya- punov functional. 1 Introduction Time-delay often appears in many biological, electrical engineering systems and mechanical applications, and in many cases, delay is a source of instability [5]. In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. [3, 9, 15, 10]). The stability issue of systems with delay is, therefore, of theoretical and practical importance. There are only a few works on Lyapunov-based technique for Partial Dif- ferential Equations (PDEs) with delay. Most of these works analyze the case of constant delays. Thus, stability conditions and exponential bounds were de- rived for some scalar heat and wave equations with constant delays and with Dirichlet boundary conditions without delay in [16, 17].

• valenciennes


• norm technique


• time-varying delay


• varying delay


• introduction time-delay


• exponential stability


• dirichlet boundary


• abstract exponential


• time dependent operator

Sujets

• Saint Nicaise
• Fridman
• Valenciennes
• Exponential stability

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 14
• Langue: English

Extrait

Stability of the heat and of the wave equations with boundary time-varying delays

Serge Nicaise ∗ , Julie Valein † , Emilia Fridman ‡

February 14, 2008

Abstract

Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be time- varying with ana priorigiven upper bound on its derivative, which is less than1. Sufficientexplicit conditions are derived that guarantee the and exponential stability. Moreover the decay rate can be explicitly computed if the data are given.

Keywordsheat equation, wave equation, time-varying delay, stability, Lya- punov functional.

1 Introduction

Time-delay often appears in many biological, electrical engineering systems and mechanical applications, and in many cases, delay is a source of instability [5]. In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. [3, 9, 15, 10]). The stability issue of systems with delay is, therefore, of theoretical and practical importance. There are only a few works on Lyapunov-based technique for Partial Dif- ferential Equations (PDEs) with delay. Most of these works analyze the case ofconstant delays stability conditions and exponential bounds were de-. Thus, rived for some scalar heat and wave equations with constant delays and with Dirichlet boundary conditions without delay in [16, 17]. Stability and instability conditions for the wave equations with constant delay can be found in [10, 12].

∗ de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Insti-Université tut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France, [e-mail] † de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Insti-Université tut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France, [e-mail] ‡ School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978 Israel, E-mail: [e-mail]

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