Reachability Analysis of Randomly Perturbed Hamiltonian Systems

In this paper, we revisit energy-based concepts of controllability and reformulate them for control-affine nonlinear systems perturbed by white noise. Specifically, we discuss the relation between controllability of deterministic systems and the corresponding stochastic control systems in the limit of small noise and in the case in which the target state is a measurable subset of the state space. We derive computable expression for hitting probabilities and mean first hitting times in terms of empirical Gramians, when the dynamics is given by a Hamiltonian system perturbed by dissipation and noise, and provide an easily computable expression for the corresponding controllability function as a function of a subset of the state variables.