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

Serge Nicaise; Julie Valein; Emilia Fridman

doi:10.3934/dcdss.2009.2.559

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2009, Volume 2, Issue 3: 559-581. Doi: 10.3934/dcdss.2009.2.559

This issue Previous Article Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation Next Article On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms

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

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

Received: March 31, 2008

Revised: August 31, 2008

Published: August 2009

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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.

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Mathematics Subject Classification: Primary: 93D15; Secondary: 35L05.

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

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