Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid–structure interaction

We consider a Bolza boundary control problem involving a fluid–structure interaction model. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The model considered consists of a linearized Navier–Stokes equation coupled on the interface with a dynamic wave equation. The model incorporates convective terms resulting from the linearization of the Navier–Stokes equation around equilibrium. The existence of the optimal control and its feedback characterization via a solution to a Riccati equation is established. The main mathematical difficulty of the problem is caused by unbounded action of control forces which, in turn, result in Riccati equations with unbounded coefficients and in singular behavior of the gain operator. This class of problems has been recently studied via the so-called Singular Estimate Control Systems (SECS) theory, which is based on the validity of the Singular Estimate (SE) [G. Avalos, Differential Riccati equations for the active control of a problem in structural acoustic, J. Optim. Theory, Appl. 91 (1996) 695–728; I. Lasiecka, Mathematical Control Theory of Coupled PDE’s, in: NSF- CMBS Lecture Notes, SIAM, 2002. with Unbounded Controls; I. Lasiecka, A. Tuffaha, Riccati Equations for the Bolza Problem arising in boundary/point control problems governed by c0 semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008) 229–246]. It is shown that the fluid–structure interaction does satisfy the Singular Estimate (SE) condition. This is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface. Thus, the established Singular Estimate allows for the application of recently developed general theory which, in turn, implies well-posedness of the feedback synthesis and of the associated Riccati Equation. Moreover, the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described.