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A Refinement in Čech Cohomology of Coron's Necessary Condition

Bryce Christopherson, Farhad Jafari

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.

A Refinement in Čech Cohomology of Coron's Necessary Condition

Abstract

Coron established a homological obstruction to continuous feedback stabilization of nonlinear control systems with and , showing that local asymptotic stabilizability implies the induced homomorphism satisfies , where . 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 must be a Čech cohomology -sphere and that the restriction of 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.
Paper Structure (5 sections, 8 theorems, 22 equations)

This paper contains 5 sections, 8 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Coron's Condition
  3. Reframing in Čech Cohomology
  4. The Strong Version of Coron's Necessary Condition
  5. Conclusion

Key Result

Theorem 2

For $k \geq 0$, the system $\dot{x}=f(x,u)$ with $f \in C(\Omega,\mathbb{R}^n)$ and $f(0,0)=0$ is locally asymptotically (resp. exponentially) stabilizable by means of a $C^{k}$ feedback law $u$ if and only if there exists a $C^{k}$ local section $\alpha$ of $f$ at the origin such that $\textrm{proj

Theorems & Definitions (15)

  • Definition 1: Local Asymptotic Stabilizability - Feedback Laws
  • Definition 2: Sections, Local Sections
  • Theorem 2
  • Theorem 3: Coron's Condition
  • Example 1: Insufficiency of \ref{['coroncondition']} coron2
  • Theorem 4: Vietoris-Begle Mapping vietoris1927hoheren
  • Theorem 5: Begle kryszewski1996remarkscalcut2012fundamental
  • Lemma 6: borsuk1967theory
  • Lemma 7: bredon1997sheaf
  • Lemma 8
  • ...and 5 more