Coron必要条件的Čech上同调精细化
A Refinement in Čech Cohomology of Coron's Necessary Condition
AI总结：
利用Čech上同调和Vietoris-Begle映射定理，将Coron关于非线性控制系统连续反馈镇定的同调障碍条件从最高类约束加强为完全上同调刚性。
摘要：
Coron建立了非线性控制系统 $\dot{x}=f(x,u)$（其中 $f \in C(\Omega,\mathbb{R}^n)$ 且 $f(0,0)=0$）连续反馈镇定的同调障碍，表明局部渐近可镇定意味着诱导同态 $f_*$ 满足 $f_*\big(H_{n-1}(\Sigma_\varepsilon)\big)=H_{n-1}(S^{n-1})$，其中 $\Sigma_\varepsilon:=\Big(\big(\mathbb{B}_\varepsilon^{\mathbb{R}^n}(0)\times\mathbb{B}_\varepsilon^{\mathbb{R}^m}(0)\big)\cap \Omega\Big)\setminus f^{-1}(0)$。本文利用Čech上同调和Vietoris-Begle映射定理精细化Coron的必要条件。具体地，我们证明 $\Sigma_\varepsilon$ 的闭包必须是Čech上同调 $(n-1)$-球面，且 $f$ 在该子集上的限制诱导所有度数上的Čech上同调群同构。这使Coron的条件从最高类的约束加强为完全上同调刚性陈述。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
