Abstract
A notion of $L^p$ -exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. 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.
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