Control against large deviation for oscillatory systems

The problem of controlling a near-Hamiltonian noisy system so as to keep it within a domain of bounded oscillations has been studied intensively in the last decade. This paper considers a new class of problems associated with control against large deviation in a weakly perturbed system. An exponential risk-sensitive residence time criterion is introduced as a performance measure, and a related HJB equation is constructed. An averaging procedure is developed for deriving an approximate solution of the risk-sensitive control problem in the small noise limit. It is shown that the averaged HJB equation is reduced to a first order PDE with the coefficients dependent on the noise intensity in the leading order term, though this intensity tends to zero in the original system. Near optimal control is constructed as a nonlinear time-independent feedback with parameters dependent on the noise intensity in the small noise limit. An example illustrates an application of this method to a system with resonance dynamics and with non-white noise perturbations.