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Local null controllability of a two-dimensional fluid-structure interaction problem. (English) Zbl 1149.35068

In the paper some controllability result for a fluid-structure interaction problem is proved. In two dimensions, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. It is proved that, for small initial data, the system under consideration is null controllable, that is, for a given \(T > 0\), the system can be driven at rest and the structure to its reference configuration at time \(T\). To show this result a linearized system is considered. Thanks to an observability of an inequality obtained from the Carleman one, it is proved an optimal controllability result with a regular control. Next, with the help of Kakutani’s fixed point theorem and regularity result the nonlinear problem is investigated. It is worth to mention that a simultaneous and independent work on a close problem has been published by O. Yu. Imanuvilov and T. Takahashi [Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9) 87, No.4, 408–437 (2007; Zbl 1124.35056)]. In these papers the methods used are different.

MSC:

35Q30 Navier-Stokes equations
76N25 Flow control and optimization for compressible fluids and gas dynamics
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations

Citations:

Zbl 1124.35056
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References:

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