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  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  in the case of the Navier-Stokes equations and abstracted in [5, 6].
Springer Proceedings in Mathematics and Statistics
Lasiecka, I., & Triggiani, R. (2015). Uniform stabilization with arbitrary decay rates of the Oseen equation by Finite-Dimensional tangential localized interior and boundary controls. Springer Proceedings in Mathematics and Statistics, 113, 125-154. https://doi.org/10.1007/978-3-319-12145-1_8