Boundary feedback control in Fluid-Structure Interactions

We consider a boundary control system for a fluid structure interaction model. This system describes the motion of an elastic structure inside a viscous fluid with interaction taking place at the boundary of the structure, and with the possibility of controlling the dynamics from this boundary. Our aim is to construct a real time feedback control based on a solution to a Riccati equation. The difficulty of the problem under study is due to the unboundedness of the control action, which is typical in boundary control problems. However, this class of unbounded control systems, due to its physical relevance, has attracted a lot of attention in recent literature (cf. [5], [18], [11]). It is known that Riccati feedback (unbounded) controls may develop strong singularities which destroy the well-posedness of Riccati equations. This makes computational implementations problematic, to say the least. However, as shown recently, this pathology does not happen for certain classes of unbounded control systems usually referred to as singular estimate control systems (SECS) (cf. [11], [21]). For such systems, there is a full and optimal Riccati theory in place, which leads to the well-posedness of feedback dynamics. Our objective is to show that the boundary control problem in question falls in the class of singular estimate control systems (SECS). Once this is accomplished, an application of the theory in [21] leads to the main result of this paper which is well-posedness of Riccati equations and of the Riccati feedback synthesis.