Min-max game theory and non-standard differential Riccati equations under the singular estimates and an application to the fluid-structure interaction model

We consider an abstract dynamical system which is characterized by "singular estimates", as it arises from many concrete hyperbolic/parabolic coupled PDE models. subject to boundary/point control and (deterministic) disturbance. For such a system, we study a min-max game theory problem with quadratic cost functional over a finite horizon. The problem is fully solved in feedback form via a Riccati operator which satisfies a non-standard differential Riccati equation.;Next, we study an application of the above theory. We consider an established hyperbolic-parabolic fluid-structure interaction model with control acting at the interface between the two media and disturbance inside the media. The structure is immersed in a fluid. The main mathematical difficulty is that such model fails to satisfy the "singular estimate" from control to state. This is a critical obstacle, as this is precisely the foundational property used in the development of the full min-max theory problem in Chapter 2-4 to include solvability of the Differential Riccati equation. Failure of the "singular estimate" property is due to a mismatch between the parabolic and hyperbolic component of the overall coupled dynamics. By introducing suitable observation or output operators, with incremental smoothness on the trajectory, it is shown that the resulting system satisfies a modified singular estimate, which is from the control to the observation space. This then allows one to adapt (and generalize) the complete min-max theory to the present fluid-structure interaction model. The approach followed is based on an abstract setting.