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Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers
We study the boundary stabilization of the two-dimensional
Navier-Stokes equations about an unstable stationary solution
by controls of finite dimension in feedback form. The main novelty
is that the linear feedback control law is determined by solving
an optimal control problem of finite dimension. More precisely,
we show that, to stabilize locally the Navier-Stokes equations,
it is sufficient to look for a boundary feedback control of
finite dimension, able to stabilize the projection of
the linearized equation onto the unstable subspace of
the linearized Navier-Stokes operator. The feedback operator
is obtained by solving an algebraic Riccati equation in a space
of finite dimension, that is to say a matrix Riccati equation.