Topological Necessary Conditions for Control Dynamics

TL;DR

This paper investigates topological necessary conditions for control systems with specific attractor types, utilizing Hopf classification and homology to understand the constraints on control dynamics.

Contribution

It introduces a topological framework using Hopf classification and homology to analyze necessary conditions for controlled dynamics with specified attractors.

Findings

Degree invariance under linear homotopy for Lyapunov level sets

Topological constraints on control dynamics derived from homology considerations

Analysis of cases where the degree is zero and surjectivity fails

Abstract

An analysis of necessary conditions for the existence of controlled dynamics with an attractor of a specified topological type is given. It uses the Hopf classification by degree for Gauss maps of manifolds to spheres of the same dimension, in this case noting that by a linear homotopy, the degree is the same for the negative of the gradient flow of a Lyapunov function and for the stabilising vector field. The manifolds are Lyapunov level sets. More general contexts are given by considering maps in homology. Some account is taken of cases where the degree is zero and the surjectivity fails. A version of this work was submitted for publication in 2009, and the paper is now in the process of revision and expansion.