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.

I. Introduction

Stationary statistics of the dynamical systems can be found as the steady-state solution of a relevant PDE with the Fokker–Plank operator [1]. However, it is difficult to solve for the Fokker–Plank equation over a finite time interval. The appropriate mathematical basis to examine the escape processes in the weakly perturbed systems is large deviations theory, which provides an alternative approach to the analysis of the weakly perturbed dynamics, opposed to the PDE solution. In this paper, we use the essentials of large deviations theory in the form developed in [2]–[5].