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Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. (English) Zbl 0983.93021

The system is described by the Navier-Stokes equations \[ \begin{aligned} &{\partial v(t, x) \over \partial t} - \Delta v(t, x) + (v(t, x), \nabla) v(t, x) + \nabla p(t, x) = f(x), \\ &\operatorname{div} v(t, x) = 0 , \quad v(0, x) = v_0(x) \end{aligned} \tag{1} \] in a two-dimensional bounded domain \(\Omega\) and \(\widehat v(x)\) is a steady-state solution of (1) satisfying the Dirichlet boundary condition. It is assumed that \(\widehat v(x)\) is an unstable singular point of the system defined by (1) and the Dirichlet boundary condition, and the object is to stabilize the system by means of a boundary control acting as follows: \[ v(t, x) |_{\partial \Omega} = u(t, x) . \] Stabilization (starting from an initial condition \(v_0\) close to \(\widehat v)\) means the construction of a control \(u\) such that \[ \|v(t, \cdot) - \widehat v(\cdot)\|_{H^1(\Omega)} \leq C e^{-\sigma t} \quad \text{as }t \to \infty \] with \(\sigma > 0.\) The author shows that this can be achieved by a control concentrated in part of the boundary \(\partial \Omega\) and given by a feedback law.

MSC:

93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
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