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May 2018, 23(3): 1107-1132. doi: 10.3934/dcdsb.2018144

## 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

Received  March 2017 Revised  August 2017 Published  February 2018

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

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.

Citation: Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144
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A 2D sample of an admissible control volume $\mathcal{D}$