Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyperviscosities

This paper deals with the optimal control of space-time statistical behavio r of turbulent fields. We provide a unified treatment of optimal control pr oblems for the deterministic and stochastic Navier-Stokes equation with lin ear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty-Browder technique to prove t he existence-bf optimal controls. For the stochastic case with monotone vis cosity, we combine the Minty-Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal contr ols. The deterministic models given in this paper also cover some simple ed dy viscosity type turbulence closure models.