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.

Keywords:

• Fluid-structure interaction
• viscoelastic damping
• uniform stability
• multipliers
• non-trivial equilibrium
• Navier–Stokes equation
• wave equation

AMS Subject Classifications:

• 35B35
• 74F10
• 93D15

Acknowledgements

The author would like to thank Professors Irena Lasiecka and Roberto Triggiani and anonymous referees for their valuable insights and feedback that have helped the author improve the article.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was partially supported by the National Science Foundation [grant number HRD-1601127].