# Exponential Consensus of Multiple Agents over Dynamic Network Topology: Controllability, Connectivity, and Compactness

## TL;DR

This paper establishes necessary and sufficient conditions for achieving exponentially fast consensus among agents with linear dynamics over dynamic networks, emphasizing controllability, network connectivity, and a new synchronization index.

## Contribution

It introduces a precompactness assumption on network topology and a synchronization index, providing a comprehensive framework for exponential consensus in dynamic networks.

## Key findings

- Controllability of agents is necessary for consensus.
- Precompactness of network Laplacian functions generalizes existing assumptions.
- Joint $(oldsymbol{ ext{δ}},T)$-connectivity and controllability ensure exponential consensus.

## Abstract

This paper investigates the problem of securing exponentially fast consensus (exponential consensus for short) for identical agents with finite-dimensional linear system dynamics over dynamic network topology. Our aim is to find the weakest possible conditions that guarantee exponentially fast consensus using a Lyapunov function consisting of a sum of terms of the same functional form. We first investigate necessary conditions, starting by examining the system (both agent and network) parameters. It is found that controllability of the linear agents is necessary for reaching consensus. Then, to work out necessary conditions incorporating the network topology, we construct a set of Laplacian matrix-valued functions. The precompactness of this set of functions is shown to be a significant generalization of existing assumptions on network topology, including the common assumption that the edge weights are bounded piecewise constant functions or continuous functions. With the aid of such a precompactness assumption and restricting the Lyapunov function to one consisting of a sum of terms of the same functional form, we prove that a joint $(\delta, T)$-connectivity condition on the network topology is necessary for exponential consensus. Finally, we investigate how the above two ``necessities'' work together to guarantee exponential consensus. To partially address this problem, we define a synchronization index to characterize the interplay between agent parameters and network topology. Based on this notion, it is shown that by designing a proper feedback matrix and under the precompactness assumption, exponential consensus can be reached globally and uniformly if the joint $(\delta,T)$-connectivity and controllability conditions are satisfied, and the synchronization index is not less than one.