Stochastic approximations of hybrid systems

This paper introduces a method for approximating the dynamics of deterministic hybrid systems. Within this setting, we shall consider jump conditions that are characterized by spatial guards. After defining proper penalty functions along these deterministic guards, corresponding probabilistic intensities are introduced and the deterministic dynamics are approximated by the stochastic evolution of a continuous-time Markov process. We would illustrate how the definition of the stochastic barriers can avoid ill-posed events such as "grazing", and show how the probabilistic guards can be helpful in addressing the problem of event detection. Furthermore, this method represents a very general technique for handling Zeno phenomena; it provides a universal way to regularize a hybrid system. Simulations would show that the stochastic approximation of a hybrid system is accurate, while being able to handle \´\´pathological cases". Finally, further generalizations of this approach are motivated and discussed.