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Feedback stabilization for the 2D Navier-Stokes equations. (English) Zbl 0997.93049
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 223, 179-196 (2002).
The system is described by the 2-dimensional Navier-Stokes equations for the velocity \(v(t, x)\) and the pressure \(p(t, x),\) \[ {\partial v(t, x) \over \partial t} - \Delta v(t, x) + (v(t, x), \nabla) v(t, x) + \nabla p(t, x) = f(x), \quad \text{div} v(t,x) = 0 \] in a two-dimensional domain \(\Omega\) with boundary \(\Gamma\) and with initial and boundary conditions \[ v(0, x) = v_0(x)\;\text{on} \Omega , \qquad v(t, x) = 0\;\text{on} \Sigma_0, \quad v(t, x) = u(t, x)\;\text{on} \Sigma \quad \] where \(\Sigma_0, \Sigma\) is a partition of \((0, \infty) \times \Gamma\). The equations have a steady state solution \((\overline v(x), \nabla \overline p(x))\), and the stabilization problem with rate \(\sigma\) is that of obtaining a control \(u(t, x)\) such that \[ \|v(t, \cdot) - \overline v(\cdot)\|_{H^1(\Omega)^2} \leq c e^{- \sigma t} \quad \text{as} t \to \infty . \] The control is required to be a feedback control according to a notion previously defined and implemented by the author for parabolic equations. The main result states that under suitable conditions, feedback stabilization with a given rate \(\sigma\) is possible.
The author presents sketches of proofs; full proofs are to be published elsewhere.
For the entire collection see [Zbl 0972.00046].

93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
35Q30 Navier-Stokes equations