Fursikov, A. V. Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. (English) Zbl 0983.93021 J. Math. Fluid Mech. 3, No. 3, 259-301 (2001). 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. Reviewer: Hector O.Fattorini (Los Angeles) Cited in 3 ReviewsCited in 45 Documents 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 Keywords:Navier-Stokes equations; Oseen equations; boundary control; feedback control; stabilization; two-dimensional bounded domain PDFBibTeX XMLCite \textit{A. V. Fursikov}, J. Math. Fluid Mech. 3, No. 3, 259--301 (2001; Zbl 0983.93021) Full Text: DOI