Control for a Weakly Perturbed Stochastic Oscillator with a Guaranteed Lifetime
A problem of controlling a noisy oscillatory system so as to prevent it from leaving a prescribed domain covers a wide range of applications. In this paper, in contrast to the great majority of control approaches, we suggest a control strategy aimed at building a system in which the escape rate and/or escape probability are independent of noise (in the small noise limit). An explicit formula for feedback control is derived. Our results exploit the properties of the Euler-Lagrange equations of motion. We demonstrate that for Lagrangian systems, in contrast to the great majority of nonlinear problems, one can construct a closed-form asymptotic solution to the first exit time problem. An explicit formula allows choosing the parameters of a regulator guaranteeing weak dependence of the escape rate on noise strength. An application of this result to the problem of trapping a particle in the betatron accelerator illustrates the theory.