Let vˆ be a velocity vector field of steady-state fluid flow in a bounded container. We do not suppose that vˆ is stable. For each fluid flow which is close to vˆ at time moment t = 0 we propose a mathematical construction of feedback control from the boundary of the container which stabilize to vˆ this flow, i.e. which forces this flow to tend to vˆ with prescribed exponential rate. We introduce a notion of “real process” which is an abstract analog of fluid flow or (in other version) of numerical solution of Navier-Stokes equations. Real process differs from exact solution of three-dimensional Navier-Stokes equaitons on some small fluctuatons. Alhtough construction of feedback control is based on precise solving of Navier-Stokes equations, feedback control obtained by this method can react on unpredictable fluctuations mentioned above damping them. Such construction can be useful for numerical calculation because there fluctuations appear always.
Stabilization of Solution to Navier-Stokes Equations by Feedback Control Defined on the Boundary
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Fursikov, AV. "Stabilization of Solution to Navier-Stokes Equations by Feedback Control Defined on the Boundary." Proceedings of the ASME 2002 Joint U.S.-European Fluids Engineering Division Conference. Volume 1: Fora, Parts A and B. Montreal, Quebec, Canada. July 14–18, 2002. pp. 1413-1418. ASME. https://doi.org/10.1115/FEDSM2002-31053
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