Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

Raymond, Jean-Pierre; Vanninathan, Muthusamy

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 65

We consider a fluid flow in a time dependent domain Ω f (t)=Ω∖Ω s (t) ¯⊂ℝ 3, surrounding a deformable obstacle Ω$$(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ω$$(t), t > 0. We prove that, for any arbitrary exponential decay rate ω > 0, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary ∂Ω$$(t) can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate ω > 0.

Classification : 93B52, 93C20, 93D15, 35Q30, 76D55, 76D05, 74F10
Keywords: Deformable boundary, feedback control, stabilization, Navier-Stokes equations

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