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

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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: April 2008

Revised: September 2008

Published: September 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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