Chain transitivity in generalized hybrid dynamics with application to simulation and stochastic approximation of hybrid systems

TL;DR

This paper investigates the asymptotic behavior of stochastic and discretized approximations to hybrid systems, establishing chain transitivity properties of their solutions' omega-limits in an abstract hybrid framework.

Contribution

It introduces a generalized hybrid system framework and demonstrates that discretizations and stochastic approximations preserve chain transitivity properties.

Findings

Omega-limits of solutions are internally chain transitive.

Discretizations and stochastic approximations produce mappings with chain transitive omega-limits.

Asymptotic solutions exhibit chain transitivity properties similar to true solutions.

Abstract

Asymptotic properties of discrete, stochastic approximations to hybrid systems, modeled as hybrid inclusions, are studied. First, the internal chain transitivity of omega-limits of solutions is concluded, along with other properties related to chain recurrence and transitivity. A concept of an asymptotic solution is proposed to describe any mapping that, asymptotically, resembles a solution, and for which the chain transitivity properties also turn out to hold. The mentioned developments are carried out in an abstract setting of a generalized hybrid system defined by a set of hybrid curves, each defined on a hybrid time domain, and possibly consisting of all solutions to a given hybrid inclusion. Then, more specific kinds of perturbed solutions to a hybrid inclusion are proposed and shown to include the solutions of a discretization and of a stochastic approximation to the hybrid inclusion. Consequently, appropriate discretizations and stochastic approximations of a hybrid inclusion produce mappings whose omega limits are internally chain transitive for the underlying hybrid inclusion.