Exact Controllability of Linear Stochastic Differential Equations and   Related Problems
  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. 
Yanqing Wang, Donghui Yang, Jiongmin Yong, and Zhiyong Yu
math.OC