Stabilization and Optimal Control of a 3-D Fluid-Structure Inter- Actions with a Weak Damping

We consider an interface problem consisting of a 3 D- fluid equation interacting with a 3 -D dynamic elasticity. The interface is moving according to the speed of the fluid. The PDE system is modeled by system of partial differential equations describing motion of an elastic body inside an incompressible fluid. The fluid is governed by Navier-
Stokes equation while the structure is represented by the system of dynamic elasticity with weak dissipation. The interface between the two environments undergoes oscillations which lead to moving frame configuration, the latter giving rise to a quasilinear system.

For such structures, control problems corresponding to minimization of vorticity/hydrodynamic pressure subject to constraints or minimization of drag are discussed. The problem is motivated by applica-
tions arising in bio-mechanics, aeroelasticity and industrial processes. In the presence of weak damping affecting the solid, the control-to-observation map is proved global-so that the size of the data can be chosen uniformly in time. This allows consideration of an infinite time horizon optimal control problem. The latter depends critically on the
estimates obtained in joint work with M. Ignatova, I. Kukavica and A. Tuffaha in the case of 3-D fluid structure interaction. These estimates are further developed in the study of finite horizon optimalcontrol problem in a joint work with L. Bociu and A. Tuffaha, andmost recently expanded to an infinite horizon optimization.

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