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 $f \in C(Ω,\mathbb{R}^n)$ and $f(0,0)=0$, showing that local asymptotic stabilizability implies the induced homomorphism $f_*$ satisfies $f_*\big(H_{n-1}(Σ_ε)\big)=H_{n-1}(S^{n-1})$, where $Σ_ε:=\Big(\big(\mathbb{B}_ε^{\mathbb{R}^n}(0)\times\mathbb{B}_ε^{\mathbb{R}^m}(0)\big)\cap Ω\Big)\setminus f^{-1}(0)$. 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 $Σ_ε$ 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.