Feedback stabilization of a linear hydro-elastic system

Lorena Bociu; Steven Derochers; Daniel Toundykov

doi:10.3934/dcdsb.2018144

Article Contents

2018, Volume 23, Issue 3: 1107-1132. Doi: 10.3934/dcdsb.2018144

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Feedback stabilization of a linear hydro-elastic system

1.
Department of Mathematics, North Carolina State University, Raleigh, NC 27607, USA
2.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

* Corresponding author: lvbociu@ncsu.edu
* Corresponding author: lvbociu@ncsu.edu

Received: February 28, 2017

Revised: July 31, 2017

Early access: February 2018

Published: June 2018

The first author is partially supported by NSF Grant DMS-1312801. The third author is partially supported by NSF Grant DMS-1616425.

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Abstract

It is known that the linear Stokes-Lamé system can be stabilized by a boundary feedback in the form of a dissipative velocity matching on the common interface [5]. Here we consider feedback stabilization for a generalized linear fluid-elasticity interaction, where the matching conditions on the interface incorporate the curvature of the common boundary and thus take into account the geometry of the problem. Such a coupled system is semigroup well-posed on the natural finite energy space [13], however, the system is not dissipative to begin with, which represents a key departure from the feedback control analysis in [5]. We prove that a damped version of the general linear hydro-elasticity model is exponentially stable. First, such a result is given for boundary dissipation of the form used in [5]. This proof resolves a more complex version, compared to the classical case, of the weighted energy methods, and addresses the lack of over-determination in the associated unique continuation result. The second theorem demonstrates how assumptions can be relaxed if a viscous damping is added in the interior of the solid.

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Mathematics Subject Classification: Primary: 74F10, 93D15; Secondary: 35Q74, 35Q35, 35R35, 93B18.

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Figure 1. A 2D sample of an admissible control volume $\mathcal{D}$

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