ABSTRACT
Let us consider the Navier-Stokes equation subject to a random (Gaussian) term i.e., the forcing field f has a mean value still denoted by f and a noise denoted by G. We can write (to simplify notation we use time-invariant forces) f (t ) = f (x, t) and the noise process G(t) = G(x,i) as a series dGk - Y ^ k S k ( x } t ) d w k ( t ) , where g = (gi,g2, • • • ) and w = (u>i, 102, • • • ) are regarded as ^2-valued functions. The stochastic noise process represented by g ( t ) d w ( t ] = ^kgk(x,t)dwk(t,u>) is normal distributed in H with a trace-class co-variance operator denoted by g2 = g 2 ( t ) and given by
' ( g 2 ( t ) u , v ) =
Tr(g 2(*))=
Impulse Control for Stochastic Navier-Stokes Equations 247
We interpret the stochastic Navier-Stokes equation as an Ito stochastic equation in variational form
in (0, T), with the initial condition
(u(0) ,v) = (u 0 )v) , (2.10)
for any v in the space V. A finite-dimensional (Galerkin) approximation of the stochastic Navier-Stokes equa-
tion can be defined as follows. Let {ei,C2, • • • } be a complete orthonormal system (i.e., a basis) in the Hilbert space H belonging to the space V (and L4). Denote by H^ the n-dimensional subspace of HI and V of all linear combinations of the first n elements {ei, 62, . . . , en}. Consider the following stochastic ODE in R."
