The problem of controlling a near-Hamiltonian noisy system so as to keep it within a domain of bounded oscillations is considered. An exponential risk-sensitive residence time criterion is introduced as a performance measure. An averaging procedure is developed to obtain an asymptotic solution of the optimal control problem. It is shown that the averaged HJB equation is reduced to a first order PDE with coefficients dependent on the noise intensity in the leading order term, though this intensity tends to zero in the original system. The leading order nearly optimal control is constructed as a nonlinear stationary feedback with parameters dependent on the noise intensity.