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Single input controllability of a simplified fluid-structure interaction model. (English) Zbl 1270.35259

The authors study a null-controllability problem for a simplified one dimensional boundary model for the motion of a rigid body in a viscous fluid, by one control force. The control variable is the velocity of the fluid at one boundary. The diffusion coefficient is equal to 1; these problems can be treated as parabolic problems as shown by E. Hopf in the late sixties.
The authors use spectral methods to prove the controllability and not the Carleman estimates of Y. Immanuvilov for Navier-Stokes equations: they introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients. In the article we find some important observability results which allow the authors to verify that the system is locally null-controllable for any positive time.

MSC:

35K55 Nonlinear parabolic equations
93B05 Controllability
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
93B40 Computational methods in systems theory (MSC2010)
93D15 Stabilization of systems by feedback
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