  Many natural systems exhibit phase transition where external environmental
conditions spark a shift to a new and sometimes quite different state.
Therefore, detecting the behavior of a stochastic dynamic system such as the
most probable transition pathway, has made sense.We consider stochastic dynamic
systems driven by Brownian motion. Based on variational principle and
Onsager-Machlup action functional theory, the variational problem is
transformed into deterministic optimal control problem. Different from
traditional variational method, optimal control theory can handle high
dimensional problems well. This paper describes the following three systems,
double well system driven by additive noise, Maire-Stein system driven by
multiplicative noise, and Nutrient-Phytoplankton-Nooplanktonr system.For
numerical computation, based on Pontryagin's Maximum Principle, we adopt method
of successive approximations to compute the optimal solution pathway, which is
the most probable transition pathway in the sense of Onsager - Machlup.
