# On the finiteness of accessibility test for nonlinear discrete-time systems

## TL;DR

This paper establishes finite tests for accessibility in certain classes of nonlinear discrete-time systems by determining a minimal input sequence length, with algorithms to compute this length and identify non-accessible points.

## Contribution

It introduces algorithms to compute the accessibility index and its upper bound for analytic and rational systems, enabling finite accessibility testing and analysis.

## Key findings

- Finite accessibility index can be computed for analytic and rational systems.
- Algorithms provide a method to determine non-accessible points.
- Relations between generic and pointwise accessibility are established.

## Abstract

It is shown that for two large subclasses of discrete-time nonlinear systems - analytic systems defined on a compact state space and rational systems - the minimum length $r^*$ for input sequences, called here accessibility index of the system, can be found, such that from any point $x$, system is accessible iff it is accessible for input sequences of length $r^*$. Algorithms are presented to compute $r^*$, as well as an upper bound for it, which can be computed easier, and hence provide finite tests for determination of accessibility. The algorithms also show how to construct the set of points from which the system is not accessible in any finite number of steps. Finally, some relations between generic accessibility of the system and accessibility of individual points in finite steps are given.