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Exact controllability and feedback stabilization from a boundary for the Navier-Stokes equations. (English) Zbl 1161.76467

Koumoutsakos, Petros (ed.) et al., Control of fluid flow. Berlin: Springer (ISBN 3-540-25140-5/pbk). Lecture Notes in Control and Information Sciences 330, 173-188 (2006).
Summary: For \(2D\) Navier-Stokes equations defined in a bounded domain \(\Omega\) we study stabilization of solution near a given steady-state flow \(\widehat v(x)\) by means of feedback control defined on a part \(\Gamma\) of boundary \(\partial\Omega\). New mathematical formalization of feedback notion is proposed. With its help for a prescribed number \(\sigma>0\) and for an initial condition \(v_0(x)\) placed in a small neighbourhood of \(\widehat v(x)\) a control \(u(t,x')\), \(x\in\Gamma\), is constructed such that solution \(v(t,x)\) of obtained boundary value problem for \(2D\) Navier-Stokes equations satisfies the inequality: \[ \|v(t,\cdot)-\widehat v\|_{H^1}\leq ce^{-\sigma t}\quad \text{for }t\geq 0. \]
For the entire collection see [Zbl 1089.76005].

MSC:

76D55 Flow control and optimization for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
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