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Poincaré-Hopf theorem for hybrid systems

Matthew D. Kvalheim

TL;DR

A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained and guard sets and resets are arbitrary multivalued maps (relations).

Abstract

A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth; distinct "modes" are not assumed to have constant dimension; and resets are arbitrary multivalued maps (relations).

Poincaré-Hopf theorem for hybrid systems

TL;DR

A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained and guard sets and resets are arbitrary multivalued maps (relations).

Abstract

A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth; distinct "modes" are not assumed to have constant dimension; and resets are arbitrary multivalued maps (relations).

Paper Structure

This paper contains 12 sections, 3 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Hybrid systems
  4. Motivation: stationary point separation principle
  5. Statement of results
  6. Preliminaries
  7. Standard notions
  8. Guard index and inflowing guards
  9. Proofs of Theorems \ref{['th:poinc-hopf-hybrid']} and \ref{['th:poinc-hopf-inflowing']}
  10. Proof of Theorem \ref{['th:poinc-hopf-hybrid']}
  11. Proof of Theorem \ref{['th:poinc-hopf-inflowing']}
  12. Conclusion

Key Result

Theorem 1

Let $\mathcal{G}\subset \mathcal{X}$ be compact subspaces of a smooth manifold $M$ with corners and $\varphi$ be a continuous local semiflow on $\mathcal{F}\coloneqq \mathcal{X}\setminus \mathcal{G}$ satisfying $\frac{\partial}{\partial t} \varphi |_{t=0} = \mathbf{v}|_{\mathcal{F}}$ for some locall

Figures (1)

  • Figure 1: An execution of a hybrid system $H=(\mathcal{X},\mathcal{F},\mathcal{G},\varphi,r)$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • Definition 1
  • Definition 2