Optimal Feedback Synthesis for Bolza Control Problem Arising in Linearized Fluid Structure Interaction

Bolza boundary control problem defined for linearized fluid structure interaction model is considered. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The main mathematical challenge of the problem is caused by unbounded action of control forces which, in turn, give rise to Riccati equations with unbounded coefficients and 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 so-called Singular Estimate (SE) [4, 27, 32]. It is shown that the fluid structure interaction does satisfy Singular Estimate (SE) condition. This is accomplished by showing that the maximal abstract parabolic regularity is transported, onto the wave dynamics, via hidden hyperbolic regularity of the boundary traces on the interface. The established Singular Estimate allows for the application of recently developed general theory which, in turn, implies well-posedness of feedback synthesis and of the associated Riccati Equation. Blow up rates of optimal control and of the feedback operator at the terminal time are provided.

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