On the Problem of Minimum Asymptotic Exit Rate for Stochastically Perturbed Multi-Channel Dynamical Systems

We consider the problem of minimizing the asymptotic exit rate with which the controlled-diffusion process of a stochastically perturbed multi-channel dynamical system exits from a given bounded open domain. In particular, for a class of admissible bounded linear feedback operators, we establish a connection between the asymptotic exit rate with which such a controlled-diffusion process exits from the given domain and the asymptotic behavior (i.e., a probabilistic characterization) of the principal eigenvalue of the infinitesimal generator, which corresponds to the stochastically perturbed dynamical system, with zero boundary conditions on the given domain. Finally, we briefly remark on the implication of our result for evaluating the performance of the associated deterministic multi-channel dynamical system, when such a dynamical system is composed with a set of (sub)-optimal admissible linear feedback operators.