Obstructions to asymptotic stabilization

TL;DR

This paper investigates the fundamental topological obstructions to achieving asymptotic stabilization in dynamical and control systems, establishing necessary conditions based on homotopy and homology theories.

Contribution

It introduces new topological necessary conditions for asymptotic stability and stabilizability, linking control theory with algebraic topology.

Findings

Homotopical necessary condition for stability

Homological necessary condition derived from the homotopical condition

Certain extensions are shown to be impossible

Abstract

Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in turn used to obtain a homological one. A certain extension is ruled out. Questions are posed.