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Applications of Lefschetz numbers in control theory

Peter Saveliev

TL;DR

This work develops a topological framework for control problems by applying Lefschetz coincidence theory to detect equilibria and assess controllability under perturbations. By generalizing the Lefschetz number to the Lefschetz homomorphism $\Lambda_{fg}$, the author derives conditions guaranteeing the existence of equilibria and controllability even when the state-input spaces have different dimensions, and provides robustness results against small or arbitrary perturbations. The paper also introduces removability criteria for coincidences and develops both discrete-time and continuous-time formulations, including practical one-step criteria and endpoint-map approaches. Collectively, these results offer a principled, homology-based method to guarantee key control properties and their persistence under perturbations. The work bridges algebraic topology and control theory, enabling rigorous analysis of equilibria, controllability, and robustness using topological invariants.

Abstract

We develop some applications of techniques of the Lefschetz coincidence theory in control theory. The topics are existence of equilibria and their robustness, controllability and its robustness.

Applications of Lefschetz numbers in control theory

TL;DR

This work develops a topological framework for control problems by applying Lefschetz coincidence theory to detect equilibria and assess controllability under perturbations. By generalizing the Lefschetz number to the Lefschetz homomorphism , the author derives conditions guaranteeing the existence of equilibria and controllability even when the state-input spaces have different dimensions, and provides robustness results against small or arbitrary perturbations. The paper also introduces removability criteria for coincidences and develops both discrete-time and continuous-time formulations, including practical one-step criteria and endpoint-map approaches. Collectively, these results offer a principled, homology-based method to guarantee key control properties and their persistence under perturbations. The work bridges algebraic topology and control theory, enabling rigorous analysis of equilibria, controllability, and robustness using topological invariants.

Abstract

We develop some applications of techniques of the Lefschetz coincidence theory in control theory. The topics are existence of equilibria and their robustness, controllability and its robustness.

Paper Structure

This paper contains 8 sections, 14 theorems, 33 equations.

Table of Contents

  1. Introduction.
  2. Preliminaries from algebraic topology.
  3. Review of Lefschetz theory.
  4. Detecting coincidences.
  5. Removing coincidences.
  6. Existence of equilibria.
  7. Controllability.
  8. Continuous systems.

Key Result

Proposition 4.3

If the degree $m$ of $h$ is zero, $L(h)=\sum_{k}(-1)^{k}Trace(h_{k})$.

Theorems & Definitions (27)

  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3
  • Theorem 4.4
  • Corollary 4.5
  • proof
  • Corollary 4.6
  • proof
  • Theorem 5.1
  • proof
  • ...and 17 more