Exponential escape criteria for a quasiconservative system with small noise

The problem of controlling a near-Hamiltonian noisy system so as to prevent it from overcoming the potential barrier is considered. An exponential risk-sensitive residence time criterion is examined as a solution of a related HJB equation. The averaged HJB equation is constructed as 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. The leading order nearly optimal control is constructed as a stationary feedback with parameters dependent on the noise intensity.