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

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September 2009, 2(3): 559-581. doi: 10.3934/dcdss.2009.2.559

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

Serge Nicaise ^1, , Julie Valein ^2, and Emilia Fridman ^3,

1.	Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9
2.	Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9
3.	School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

Received April 2008 Revised September 2008 Published June 2009

Abstract
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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: stability, Heat equation, wave equation, time-varying delay, Lyapunov functional..

Mathematics Subject Classification: Primary: 93D15; Secondary: 35L0.

Citation: Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

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