Rapid stabilization for Stokes and Navier-Stokes systems with localized disturbance

Abstract

In this study, we analyze the rapid stabilization of Stokes and the two- and three-dimensional Navier-Stokes systems in the presence of an unknown spatially localized disturbance. A multivalued feedback control strategy is proposed which combines the Lyapunov frequency method with the multivalued sign operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator provides robustness against disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach guarantees global rapid exponential stabilization for the Stokes system and local rapid exponential stabilization for the two- and three-dimensional Navier-Stokes systems, with arbitrarily large disturbance bounds. In three dimensions, we obtain global strong solutions and rapid exponential stabilization for initial data sufficiently small in the $L^2$-norm with a prescribed, arbitrarily large bound in the $H^1$-norm.