A Refinement in Čech Cohomology of Coron's Necessary Condition

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 $f_{\ast }$ satisfies $f_{\ast }(H_{n-1}({\Sigma }_{\epsilon }))=H_{n-1}(S^{n-1})$, where ${\Sigma }_{\epsilon }:=\left(\right({\mathbb{B}}_{\epsilon }^{{\mathbb{R}}^n}(0)\times {\mathbb{B}}_{\epsilon }^{{\mathbb{R}}^m}(0))\cap \Omega )\setminus f^{-1}(0)$. In this paper, we refine Coron's necessary condition using Čech cohomology and the Vietoris-Begle mapping theorem. Specifically, we prove that the closed version of ${\Sigma }_{\epsilon }$ must be a Čech cohomology $(n-1)$-sphere and that the restriction of $f$ to this subset induces an isomorphism on its Čech cohomology groups in all degrees. This strengthens Coron's condition from a constraint on the top class to a full cohomological rigidity statement.

Withdrawn by the authors because the manuscript relies on a supporting result that requires correction; consequently, the main argument is not justified in its present form. The present version should not be cited