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Consensus under Persistence Excitation

Fabio Ancona, Mohamed Bentaibi, Francesco Rossi

Abstract

We prove that a first-order cooperative system of interacting agents converges to consensus if the so-called Persistence Excitation condition holds. This condition requires that the interaction function between any pair of agents satisfies an integral lower bound. The interpretation is that the interaction needs to ensure a minimal amount of service.

Consensus under Persistence Excitation

Abstract

We prove that a first-order cooperative system of interacting agents converges to consensus if the so-called Persistence Excitation condition holds. This condition requires that the interaction function between any pair of agents satisfies an integral lower bound. The interpretation is that the interaction needs to ensure a minimal amount of service.
Paper Structure (9 sections, 5 theorems, 50 equations, 2 figures)

This paper contains 9 sections, 5 theorems, 50 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Models of opinion formation
  3. General properties
  4. Dynamics on the real line
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof in $\mathbb{R}$
  7. Proof for any dimension
  8. Simulations
  9. Conclusions and future directions

Key Result

Theorem I.1

Let $\{x_i(t)\}_{i=1}^N$ be a solution of e-ODE with initial data $\{\bar{x}_i\}_{i=1}^N$. Assume the following conditions: Fix $T,\mu>0$ and assume that all $M_{ij}$ satisfy defof:PEgeneral. Then, consensus holds:

Figures (2)

  • Figure 1: Solutions of \ref{['e-ODE']} with varying $\mu$.
  • Figure 2: The average time to convergence as a function of $\mu$.

Theorems & Definitions (6)

  • Definition I.1: Persistent excitation
  • Theorem I.1
  • Lemma II.1
  • Lemma II.2
  • Proposition II.1
  • Lemma II.3