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Equivalence between observability and stabilization for a class of second order semilinear evolution

Pages: 744 - 752, Issue Special, September 2009

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Louis Tcheugoue Tebou - Department of Mathematics, Florida International University, University Park, Miami, Florida 33199, United States (email)

Abstract: We consider an abstract second order semilinear evolution equation with a bounded dissipation. We establish an equivalence between the stabilization of this system and the observability of the corresponding undamped system. Our technique of proof relies on an appropriate decomposition of the solution, and the energy method. Our result generalizes an earlier one by Haraux [5] who studied the same type of problem for linear systems. Some applications of our result are provided, and the paper ends with a few open problems.

Keywords:  second order evolution equation, semilinear equation, hyperbolic equation, observability, stabilization, bounded dissipation, localized damping, Euler-Bernoulli equation, elasticity equations
Mathematics Subject Classification:  Primary: 93D15; Secondary: 35L70, 35L05, 37L15

Received: August 2008;      Revised: February 2009;      Published: September 2009.