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

Rafal K. Goebel; Andrew R. Teel

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arXiv · 2604.08807

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

Rafal K. Goebel·Andrew R. Teel

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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.

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BibTeX RIS

Rafal K. Goebel, Andrew R. Teel. 2026-04-09. Chain transitivity in generalized hybrid dynamics with application to simulation and stochastic approximation of hybrid systems. https://arxiv.org/abs/2604.08807

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