# Finite determination of accessibility and geometric structure of singular points for nonlinear systems

## TL;DR

This paper uses algebraic geometry to analyze the finiteness and geometric structure of singular points related to accessibility in nonlinear polynomial and analytic systems, providing algorithms and bounds for these properties.

## Contribution

It introduces constructive methods and algorithms to determine the maximum Lie bracket depth for accessibility and identifies all singular points, advancing understanding of system non-holonomy.

## Key findings

- Algorithms for maximum Lie bracket depth
- Upper bounds on accessibility index
- Complete set of singular points identified

## Abstract

Exploiting tools from algebraic geometry, the problem of finiteness of determination of accessibility/strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems. The results are constructive, and algorithms are given to find the maximum depth of Lie brackets necessary for deciding accessibility/strong accessibility of the system at any point, called here accessibility/strong accessibility index of the system, and is known as the degree of non-holonomy in the literature. Alternatively, upper bounds on the accessibility/strong accessibility index are obtained, which can be computed easier. In each approach, the entire set of accessibility/strong accessibility singular points are obtained.