Appl Math Optim 41:255–308 (2000)
2000 Springer-Verlag New York Inc.
Deterministic and Stochastic Control of Navier–Stokes Equation
with Linear, Monotone, and Hyperviscosities
S. S. Sritharan
Code D73H, Space and Naval Warfare Systems Center,
San Diego, CA 92152, USA
Abstract. This paper deals with the optimal control of space–time statistical be-
havior of turbulent ﬁelds. We provide a uniﬁed treatment of optimal control prob-
lems 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 of Stroock and Varadhan to es-
tablish existence of optimal controls. The deterministic models given in this paper
also cover some simple eddy viscosity type turbulence closure models.
Key Words. Stochastic control, Control of ﬂuids, Turbulence control, Navier–
Stokes equation, Minty–Browder theory.
AMS Classiﬁcation. 76, 49, 60.
Optimal control theory of viscous ﬂow has been a rapidly developing subject during the
past several years . Most of the work is concerned with the case of deterministic con-
trol. Stochastic dynamic programming is developed in  by the author and nonlinear
stochastic ﬁltering is developed in . In this paper we obtain the existence of optimal
controls for stochastically forced ﬂuid ﬂow with Newtonian and non-Newtonian consti-
This research was supported by the Ofﬁce of Naval Research.