ABSTRACT
Controlling fluid turbulence is one of the most important scientific problems today. During the past several years major mathematical developments have taken place in optimal control theory of viscous time dependent fluid dynamics [7]. This paper deals with statistical properties of turbulence under control action. We show that it is possible to choose a stationary control that corresponds to a statistically stationary turbulent state with certain prescribed statistical moment attaining a minimum. This type of mathematical result has practical significance in the area of Reynolds stress closure modeling for controlled turbulent flows. *
Let G be a bounded open domain in R 2 with a smooth boundary 3G. Let T be an arbitrary but fixed positive number. For t є [0, T], consider
(24.1.1)
and
with
Here p denotes the pressure field and is a scalar-valued function. W is the generalized derivative of an Я -valued Wiener process W, where Я is a suitable Hilbert space. W has a nuclear covariance form Q on H . g : Я -> L q (H) where,
for all h є Я and {*?,·} CONS in Я. v is the control which takes values in some metrizable Lusin space U. N is a linear or
non-linear operator representing possible non-linearities in the actuator term. Exogeneous forces such as structural vibrations and other body, forces are modeled by the noise term.
