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Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyperviscosities. (English) Zbl 0951.76023
Summary: This paper deals with the optimal control of space-time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier-Stokes equation with linear 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 the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty-Browder technique with the martingale problem formulation to establish the existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models.

MSC:
76D55 Flow control and optimization for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
49J20 Existence theories for optimal control problems involving partial differential equations
49J55 Existence of optimal solutions to problems involving randomness
76M35 Stochastic analysis applied to problems in fluid mechanics
76F55 Statistical turbulence modeling
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