  A notion of $L^p$-exact controllability is introduced for linear controlled
(forward) stochastic differential equations, for which several sufficient
conditions are established. Further, it is proved that the $L^p$-exact
controllability, the validity of an observability inequality for the adjoint
equation, the solvability of an optimization problem, and the solvability of an
$L^p$-type norm optimal control problem are all equivalent.
