Lyapunov controllability and global optimal control

Certain problems in nonlinear optimal control where we are required to steer the state away from attracting sets are related to global Lyapunov functions for the uncontrolled dynamics. In particular. the optimal cost as a function of terminal point, starting from an attracting set turns out to be a Lyapunov function. The geometric condition that gives this result is a kind of controllability that relates the control dynamics and the gradient of the Lyapunav function. This problem, arises in large deviation theory for diffusions and its solution yields new qualitative results for the exit paths of large deviations.