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Exponential convergence of multiagent systems with lack of connection

Fabio Ancona, Mohamed Bentaibi, Francesco Rossi

TL;DR

The paper tackles consensus and flocking in multi-agent networks with intermittent connections by leveraging Moreau's $(T,\mu)$-connectivity and introducing the graph length $d^*$ as a key quantitative measure. It provides explicit exponential convergence rates for first-order linear dynamics and extends the results to second-order systems, yielding alignment and flocking criteria that depend on the decay properties of the interaction function $\phi$ and the connectivity length. The main methodology combines barrier arguments with graph-based estimates and reduces nonlinear dynamics to a one-dimensional linear framework, enabling explicit constants and time scales. The work also derives practical corollaries under persistent excitation and integral scrambling coefficients, and contrasts these results with existing symmetric-consensus literature, highlighting the benefits of directed connectivity in guaranteeing convergence with measurable rates.

Abstract

Finding conditions ensuring consensus, i.e. convergence to a common value, for a networked system is of crucial interest, both for theoretical reasons and applications. This goal is harder to achieve when connections between agents are temporarily lost. Here, we prove that known conditions (introduced by Moreau) ensure an exponential convergence to consensus, with explicit rate of convergence. The key result is related to the length of the graph (i.e. the number of connections to reach a common agent): if this is large, then convergence is slow. This general result also provides conditions for convergence of second-order cooperative systems with lack of connections.

Exponential convergence of multiagent systems with lack of connection

TL;DR

The paper tackles consensus and flocking in multi-agent networks with intermittent connections by leveraging Moreau's -connectivity and introducing the graph length as a key quantitative measure. It provides explicit exponential convergence rates for first-order linear dynamics and extends the results to second-order systems, yielding alignment and flocking criteria that depend on the decay properties of the interaction function and the connectivity length. The main methodology combines barrier arguments with graph-based estimates and reduces nonlinear dynamics to a one-dimensional linear framework, enabling explicit constants and time scales. The work also derives practical corollaries under persistent excitation and integral scrambling coefficients, and contrasts these results with existing symmetric-consensus literature, highlighting the benefits of directed connectivity in guaranteeing convergence with measurable rates.

Abstract

Finding conditions ensuring consensus, i.e. convergence to a common value, for a networked system is of crucial interest, both for theoretical reasons and applications. This goal is harder to achieve when connections between agents are temporarily lost. Here, we prove that known conditions (introduced by Moreau) ensure an exponential convergence to consensus, with explicit rate of convergence. The key result is related to the length of the graph (i.e. the number of connections to reach a common agent): if this is large, then convergence is slow. This general result also provides conditions for convergence of second-order cooperative systems with lack of connections.

Paper Structure

This paper contains 16 sections, 13 theorems, 87 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Cooperative systems
  3. Well-posedness and contractivity of the support
  4. Reduction to 1-dimensional linear systems
  5. Models of opinion formation
  6. Proof of main result
  7. Barrier functions
  8. Estimates and connectivity graph
  9. Proof of Theorem \ref{['t-main']} and Corollary \ref{['c-main']}
  10. Additional results
  11. Persistent excitation
  12. Integral Scrambling Coefficients
  13. Flocking
  14. Examples and comparison with the literature
  15. Comparison with the literature
  16. ...and 1 more sections

Key Result

Theorem 1

Let the system e-ODE be given with $\lambda_i=\phi_{ij}\equiv 1$. Assume the following hypothesis: Fix $T,\mu>0$ and consider the $(T,\mu)$-connectivity graph $G(t)$ associated to the $M_{ij}(t)$. Assume that there exists a constant graph $G^*$ such that all its arrows are contained in $G(kT)$ for all $k\in\mathbb{N}$. Assume moreover that $G^*$ admits a globally reachable node $I$ and has length

Figures (3)

  • Figure 1: Example \ref{['ex-1']} with $N=4$.
  • Figure 2: Example \ref{['ex-2']}: interaction graph.
  • Figure 3: Example \ref{['ex-2']}: trajectories of the system for $N\in \{4,6,8\}$ with position (top) and velocity (bottom)

Theorems & Definitions (20)

  • Definition 1: $(T,\mu)$-connectivity graph
  • Definition 2
  • Theorem 1: Main result - 1st order linear systems
  • Corollary 2
  • Theorem 3: Main result - 2nd order systems
  • Remark 3: Comparison with Cucker-Smale model
  • Proposition 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • ...and 10 more