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IMA Journal of Mathematical Control and Information Advance Access originally published online on October 28, 2005
IMA Journal of Mathematical Control and Information 2006 23(2):237-257; doi:10.1093/imamci/dni056
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© The author 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Wave energy decay under fractional derivative controls

Brahima Mbodje**

Real Results Tutoring Service, 13000 F York Road PMB 176, Charlotte, NC 28278-7602, USA

** Present address: Division of Mathematics and Sciences, Rust College, 150 Rust Avenue, Holly Springs, MS 38635, USA

In this article, we investigate the asymptotic behaviour of solutions of the 1D wave equation with a boundary viscoelastic damper of the fractional derivative type. We show that the system is well-posed in the sense of semigroup. We also prove that the associated semigroup is not exponentially stable, but only strongly asymptotically so. Finally, we establish the following result. Provided that the initial states of the system are chosen sufficiently smooth and the relaxation function of the viscoelastic damper is exponentially decreasing, then solutions of the system will decay, as time goes to infinity, as

[graphic: see PDF]

A > 0.

Keywords: fractional calculus; fractional derivative controls; wave equation; energy functionals; LaSalle's invariance principle; asymptotic stability; contraction semigroup; exponential stability; stabilization.

Received on 30 November 2004.


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