Simultaneous stabilization of a system of interacting plate and membrane

Louis Tebou

doi:10.3934/eect.2013.2.153

Article Contents

2013, Volume 2, Issue 1: 153-172. Doi: 10.3934/eect.2013.2.153

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Simultaneous stabilization of a system of interacting plate and membrane

Louis Tebou^1,

1.
Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received: June 30, 2012

Revised: August 31, 2012

Published: February 2013

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Abstract

We consider a system consisting of a wave equation and a plate equation, where the rotational forces may be accounted for, in a bounded domain. The system is coupled through the dissipation which is localized in an appropriate portion of the domain under consideration. First, we show that this system is strongly stable; the feedback control region is arbitrarily small when the rotational inertia is greater than or equal to one, but it is required that its boundary intersect a nonnegligible portion of the boundary of the domain under consideration if the rotational inertia is less than one. Next we show that the system accounting for rotational forces is not exponentially stable. Afterwards, using a constructive frequency domain method, we show that the system with no rotational forces is exponentially stable provided that the feedback control region is large enough. New uniqueness and controllability results are derived.

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Mathematics Subject Classification: Primary: 93D15; Secondary: 93C20, 35L05, 37L15, 74K20.

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