## 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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