Boundary feedback stabilization of a viscous incompressible fluid in a bounded domain

Abstract

This paper presents a boundary feedback control design based on Lyapunov's direct method to globally practically exponentially stabilize a viscous incompressible fluid governed by Navier-Stokes equations at its equilibrium state in a bounded domain in three dimensional space. The control is implemented on a part of the rigid boundary and requires only boundary measurements. The Rothe method and an elliptic approximation are used to handle the time-dependent domain due to the boundary control in the proof of global existence of a weak solution of the closed-loop system. Due to consideration of less regular initial values of the fluid velocity (i.e., a global weak solution of the closed-loop system is obtained), the forces induced by the fluid on the part of the boundary, on which the control is implemented, are not able to bound. Thus, the paper derives the bound on “fluid work” on that part of the boundary in stability and convergence analysis of the closed-loop system. The advantage of considering the weak solution is its global existence and less regularity of the initial data versus local existence of strong solution requiring more regularity of the initial data.