  Recently it has been found that for a stochastic linear-quadratic optimal
control problem (LQ problem, for short) in a finite horizon, open-loop
solvability is strictly weaker than closed-loop solvability which is equivalent
to the regular solvability of the corresponding Riccati equation. Therefore,
when an LQ problem is merely open-loop solvable not closed-loop solvable, which
is possible, the usual Riccati equation approach will fail to produce a state
feedback representation of open-loop optimal controls. The objective of this
paper is to introduce and investigate the notion of weak closed-loop optimal
strategy for LQ problems so that its existence is equivalent to the open-loop
solvability of the LQ problem. Moreover, there is at least one open-loop
optimal control admitting a state feedback representation. Finally, we present
an example to illustrate the procedure for finding weak closed-loop optimal
strategies.
