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Article Dans Une Revue SIAM Journal on Control and Optimization Année : 2011

Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system

Résumé

Let $A : \mathcal{D}(A)\to \mathcal{X}$ be the generator of an analytic semigroup and $B : \mathcal{U} \to [{\cal D}(A^*)]'$ a quasi-bounded operator. In this paper, we consider the stabilization of the system $y'=Ay+Bu$ where $u$ is the linear combination of a family $(v_1,\ldots,v_K)$. Our main result shows that if $(A^*,B^*)$ satisfies a unique continuation property and if $K$ is greater or equal to the maximum of the geometric multiplicities of the the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,\ldots,v_K)$. With the same functional framework, we also prove the stabilizability of a class of nonlinear system when using feedback or dynamical controllers. We apply these results to stabilize the Navier--Stokes equations in 2D and in 3D by using boundary control.
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Dates et versions

hal-00431041 , version 1 (10-11-2009)
hal-00431041 , version 2 (08-11-2010)

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Mehdi Badra, Takéo Takahashi. Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system. SIAM Journal on Control and Optimization, 2011, 49 (2), pp.420-463. ⟨10.1137/090778146⟩. ⟨hal-00431041v2⟩
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