Completeness of Lyapunov Abstraction

TL;DR

This paper investigates the conditions under which discrete abstractions of dynamical systems are complete, revealing limitations of transversal partitioning functions and emphasizing the role of stable and unstable manifolds in abstraction completeness.

Contribution

It introduces a new approach allowing non-transversal partitioning functions and characterizes the critical points necessary for abstraction completeness.

Findings

Complete abstractions require critical points to include stable or unstable manifolds.

Transversal partitioning functions cannot produce complete abstractions.

Allowing non-positive directional derivatives complicates the abstraction process.

Abstract

In this work, we continue our study on discrete abstractions of dynamical systems. To this end, we use a family of partitioning functions to generate an abstraction. The intersection of sub-level sets of the partitioning functions defines cells, which are regarded as discrete objects. The union of cells makes up the state space of the dynamical systems. Our construction gives rise to a combinatorial object - a timed automaton. We examine sound and complete abstractions. An abstraction is said to be sound when the flow of the time automata covers the flow lines of the dynamical systems. If the dynamics of the dynamical system and the time automaton are equivalent, the abstraction is complete. The commonly accepted paradigm for partitioning functions is that they ought to be transversal to the studied vector field. We show that there is no complete partitioning with transversal functions, even for particular dynamical systems whose critical sets are isolated critical points. Therefore, we allow the directional derivative along the vector field to be non-positive in this work. This considerably complicates the abstraction technique. For understanding dynamical systems, it is vital to study stable and unstable manifolds and their intersections. These objects appear naturally in this work. Indeed, we show that for an abstraction to be complete, the set of critical points of an abstraction function shall contain either the stable or unstable manifold of the dynamical system.