Uniform stabilization with arbitrary decay rates of the Oseen equation by Finite-Dimensional tangential localized interior and boundary controls


We consider an unstable Oseen equation (linearized Navier-Stokes equations) defined on a 2-d or 3-d open connected bounded domain and subject to two types of ‘tangential’ controls: (i) a (Dirichlet-type) tangential boundary control acting on an arbitrarily small open sub-portion Γ of positive measure of the full boundary Γ; and (ii) an interior control acting tangentially on a localized collar of the boundary supported on Γ. The main result of the paper asserts, constructively, feedback uniform stabilization (even at a higher norm than L2) of the Oseen equation with arbitrary decay rates by means of a pair of controls of the type noted above, which moreover are both finite dimensional and in feedback form. The basic approach is based on the strategy introduced in [28] in 1975, though under considerably more involved technicalities, even in the analysis of the finite dimensional fully general unstable projected dynamics. Subsequent work will then yield local uniform stabilization in a neighborhood of an equilibrium (or steady-state) solution of the full Navier-Stokes model by combining the present result with techniques introduced in [4] in the case of the Navier-Stokes equations and abstracted in [5, 6].

Publication Title

Springer Proceedings in Mathematics and Statistics