Reachability Analysis of Randomly Perturbed Hamiltonian Systems

TL;DR

This paper extends energy-based controllability concepts to stochastic Hamiltonian systems with noise, deriving practical formulas for hitting probabilities, mean first hitting times, and controllability functions.

Contribution

It introduces a novel reformulation of controllability for stochastic control-affine systems and provides computable expressions for key probabilistic measures in perturbed Hamiltonian dynamics.

Findings

Derived formulas for hitting probabilities and mean first hitting times.

Provided an explicit expression for the controllability function.

Validated the approach on systems with dissipation and noise.

Abstract

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.