A Refinement in \v{C}ech Cohomology of Coron's Necessary Condition

TL;DR

This paper refines Coron's homological obstruction for stabilizing nonlinear control systems by using cech cohomology, showing the subset involved must be a cech cohomology sphere with an isomorphism induced by the system map.

Contribution

It introduces a cohomological refinement of Coron's necessary condition, strengthening the topological constraints for feedback stabilization.

Findings

The subset cech cohomology sphere condition is necessary for stabilization.

The restriction of the system map induces an isomorphism on cohomology groups.

The refinement uses cech cohomology and the Vietoris-Begle theorem to strengthen previous results.

Abstract

Coron established a homological obstruction to continuous feedback stabilization of nonlinear control systems $\dot{x}=f(x,u)$ with $f \in C(\Omega,\mathbb{R}^n)$ and $f(0,0)=0$, showing that local asymptotic stabilizability implies the induced homomorphism $f_*$ satisfies $f_*\big(H_{n-1}(\Sigma_\epsilon)\big)=H_{n-1}(S^{n-1})$, where $\Sigma_\epsilon:=\Big(\big(\mathbb{B}_\epsilon^{\mathbb{R}^n}(0)\times\mathbb{B}_\epsilon^{\mathbb{R}^m}(0)\big)\cap \Omega\Big)\setminus f^{-1}(0)$. In this paper, we refine Coron's necessary condition using \v{C}ech cohomology and the Vietoris-Begle mapping theorem. Specifically, we prove that the closed version of $\Sigma_\epsilon$ must be a \v{C}ech cohomology $(n-1)$-sphere and that the restriction of $f$ to this subset induces an isomorphism on its \v{C}ech cohomology groups in all degrees. This strengthens Coron's condition from a constraint on the top class to a full cohomological rigidity statement.