2010
DOI: 10.1137/080744761
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Feedback Stabilization of a Fluid-Structure Model

Abstract: We study a system coupling the incompressible Navier-Stokes equations in a 2D rectangular type domain with a damped Euler-Bernoulli beam equation, where the beam is a part of the upper boundary of the domain occupied by the fluid. Due to the deformation of the beam the fluid domain depends on time. We prove that this system is exponentially stabilizable, locally about the null solution, with any prescribed decay rate, by a feedback control corresponding to a force term in the beam equation. The feedback is det… Show more

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Cited by 59 publications
(73 citation statements)
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“…The advantage of this formulation is that we can prove the R-boundedness of the resolvent operator just by using the fact that Stokes operator with homogeneous Dirichlet boundary conditions is R-sectorial and a perturbation argument. This idea has been used in several fluidsolid interaction problems in the Hilbert space setting and when the structure is deformable and located at the boundary (see, for instance, [20,17] and references therein).…”
Section: R-sectoriality Of the Operator Amentioning
confidence: 99%
“…The advantage of this formulation is that we can prove the R-boundedness of the resolvent operator just by using the fact that Stokes operator with homogeneous Dirichlet boundary conditions is R-sectorial and a perturbation argument. This idea has been used in several fluidsolid interaction problems in the Hilbert space setting and when the structure is deformable and located at the boundary (see, for instance, [20,17] and references therein).…”
Section: R-sectoriality Of the Operator Amentioning
confidence: 99%
“…In fact the concept of duality between controllability and observability dates back to the celebrated Hilbert Uniqueness method (HUM), introduced in the article [25], which reduces the question of exact controllability problem of a partial differential equation into proving the observability estimate of the corresponding adjoint problem. The Carleman estimate obtained in this article can be used to prove controllability results corresponding to more complicated coupled dynamical systems, like the ones considered in [29] and [28]. In fact the two main advantages of using Carleman estimate in studying the controllability properties of a PDE are: (i) Using suitably large Carleman parameter s one can readily incorporate lower order terms especially a L ∞ potential in a linear PDE model to study the controllability.…”
Section: )mentioning
confidence: 97%
“…(3) in the domain drawn in Fig. 2, with Raymond (2010). The proof consists in showing that the infinitesimal generator of the linearized model is an analytic semigroup (when c > 0), that its resolvent is compact, and that the Hautus criterion is satisfied.…”
Section: Feedback Stabilization Of the Navier-stokes System Coupled Wmentioning
confidence: 97%