Saddle exits in optimal control problems

The purpose of this paper is to give conditions under which an optimal exit path starting from an attractor of a vector field and exiting its region of attraction passes through a saddle critical element. The control action is applied linearly with a control matrix ??(x) (not necessarily non-singular). The results thus generalize the ´mountain-pass´ type theorems that say that the ´shortest way out of a valley is from the lowest mountain pass´. After a brief review of the gradient vector field case, which we give in a global setting using Morse functions, we turn to the condition of Lyapunov controllability (see Kappos [1]) as a key to the generalization of the above results to dynamics that are dissipative.