Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions

TL;DR

This paper develops a framework for optimal boundary control of a fluid-structure interaction model involving coupled parabolic and hyperbolic PDEs, establishing well-posedness and feedback synthesis of optimal controls.

Contribution

It introduces new trace regularity results and applies a unified theory to derive optimal controls for complex PDE systems including fluid-structure interactions.

Findings

Established well-posedness of the control problem.

Derived feedback synthesis for the optimal control.

Allowed general observation functionals including energy integrals.

Abstract

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique optimal control, under a very weak constraint on the observation operator; in particular, the present analysis allows general functionals, such as the integral of the natural energy of the physical system. Furthermore, this work confirms that the theory developed in Acquistapace et al. [Adv. Differential Equations, 2005] -- crucially utilized here -- encompasses widely differing PDE problems, from thermoelastic systems to models of acoustic-structure and, now, fluid-structure interactions.