Min-Max Game Theory for the Linearized Navier-Stokes Equations with Internal Localized Control and Distributed Disturbance

We consider the abstract linearized Navier-Stokes equations with no-slip Dirichlet boundary conditions with added control and disturbance. We then study a min-max game theory problem with quadratic cost functional depending on a parameter gamma over an infinite time horizon. We demonstrate the existence of a critical value gamma_c such that: (1) if gamma > gamma_c, then the game theory problem is fully solved in feedback form via a Riccati operator which satisfies an algebraic Riccati equation (dependent on gamma) (2) if 0