  It is a longstanding unsolved problem to characterize the optimal feedback
controls for general linear quadratic optimal control problem of stochastic
evolution equation with random coefficients. A solution to this problem is
given in [21] under some assumptions which can be verified for interesting
concrete models, such as controlled stochastic wave equations, controlled
stochastic Schr\"odinger equations, etc. More precisely, the authors establish
the equivalence between the existence of optimal feedback operator and the
solvability of the corresponding operator-valued, backward stochastic Riccati
equations. However, their result cannot cover some important stochastic partial
differential equations, such as stochastic heat equations, stochastic stokes
equations, etc. A key contribution of the current work is to relax the
$C_0$-group assumption of unbounded linear operator $A$ in [21] and using
contraction semigroup assumption instead. Therefore, our result can be well
applicable in the linear quadratic problem of stochastic parabolic partial
differential equations. To this end, we introduce a suitable notion to the
aforementioned Riccati equation, and some delicate techniques which are even
new in the finite dimensional case.

Optimization and Control