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Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: application to the Navier-Stokes system. (English) Zbl 1217.93137

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

MSC:

93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
76D55 Flow control and optimization for incompressible viscous fluids
93B52 Feedback control
93C20 Control/observation systems governed by partial differential equations
35Q93 PDEs in connection with control and optimization
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