Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

Badra, Mehdi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 934-968.

We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the boundary of the domain. We determine a linear feedback law by solving a linear quadratic control problem for the linearized extended system. We show that this feedback law also stabilizes the nonlinear extended system.

Classification : 35Q30, 76D05, 76D07, 76D55, 93B52, 93C20, 93D15
Mots clés : Navier-Stokes equation, feedback stabilization, Dirichlet control, Riccati equation

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