Control of a weakly perturbed Lagrangian system with a guaranteed escape rate

This paper considers a problem of controlling a weakly perturbed Lagrangian system so as to keep it within a reference domain on a pregiven time interval. Escape from this domain is associated with failure of the system. The criterion of interest is the rate of escape, and the task is to design a controller ensuring a noise-independent escape rate (in the small noise limit) of the controlled system. We treat this problem in the context of control against large deviations in weakly perturbed dissipative systems. As this paper demonstrates, for Lagrangian systems, in contrast to the great majority of large deviation problems, an explicit logarithmic asymptotic of the escape rate can be found. An explicit formula allows us to define the parameters of a regulator guaranteeing weak dependence of the escape rate on the noise strength. The regulator consists of two units, nonlinear velocity feedback with the parameters depending on the noise strength and state feedback, independent of noise. The first unit stabilizes the system and ensures a noise-independent asymptotics of the logarithmic escape rate; the second unit is designed to modify the exit location on the boundary of the reference domain. Applications of these results to stabilization and tracking models illustrate the theory.