Stabilization of a Nonlinear Fluid Structure Interaction via Feedback Controls and Geometry of the Domain

Asymptotic stability of finite energy solutions to a fluid-structure interaction with a static interface in a bounded domain Ω∈Rnn, n=2, is considered. Nonlinear interior damping and dynamic and static boundary damping are exploited to stabilize the system. It is shown that the undamped model subject to “partial flatness” geometric condition on the interface produces solutions whose energy converge strongly to zero; while with a stress type feedback control applied on the interface of the structure, the model produces solutions whose energy is exponentially stable. An addition of a static damping on the interface produces solutions whose full norm in the phase space is exponentially stable. Without a static damping, an interesting phenomenon occurs: steady state solutions (equilibria) might generate genuinely growing in time solutions. This purely nonlinear phenomenon is captured by newly developed techniques amenable to handle instability of steady state solutions arising from nonlinearity. KeywordsFluid Strusture Interaction; Interface Control; Navier-Stokes Equation; System of Elasticity; Feedback Boundary Control; Strong Stability; Uniform Stability; Optimal Control; Passive Damping; Active Damping; Dynamic and Static Damping