Approximation of Escape Time for Lagrangian Systems With Fast Noise

This note is concerned with the large deviations asymptotics estimate of the mean escape time for Lagrangian systems subjected to fast Gaussian noise. The solution to the Hamilton-Jacobi equation of the associated variational problem is derived as a sum of two terms dependent on kinetic and potential energy of the system, respectively. A closed-form solution for classes of linear and nonlinear systems is obtained. An application to a controlled pointing problem is discussed as an example.