Global uniform stabilization to nontrivial equilibrium of a nonlinear fluid viscoelastic-structure interaction

Abstract

Uniform stability to a non-trivial equilibrium of a nonlinear fluid structure interaction model is studied. To achieve this goal, control action depending on the equilibrium and applied to the fluid is proposed. The stabilization result obtained is global and no assumptions on the smallness of the initial data or the size of equilibrium point are needed. Due to viscoelasticity, the boundary transmission conditions are highly unbounded, which requires perturbation independent argument. To overcome this difficulty, we seek to construct special multipliers based on the Stokes solver and the projection operator from to a special subspace expanded by the eigenfunction corresponding to the smallest eigenvalue of with zero Neumann boundary condition.