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Li et al. Intell Robot 2023;3(2):213-21 Intelligence & Robotics
DOI: 10.20517/ir.2023.13

Research Article Open Access

The cooperatability of the first-order multi-agent sys-
tems consisting of a leader and a follower with multi-

plicative noises under Markov switching topologies

Dianqiang Li, Tao Li

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China.

Correspondence to: Dr. Dianqiang Li, School of Mathematical Sciences, East China Normal University, No.5005, South Lianhua
Road, Shanghai 200241, China. E-mail: ldq2015ahdx@126.com.

How to cite this article: Li D, Li T. The cooperatability of the first-order multi-agent systems consisting of a leader and a follower
with multiplicative noises under Markov switching topologies. Intell Robot 2023;3(2):213-21. http://dx.doi.org/10.20517/ir.2023.13
Received: 24 Mar 2023 First Decision: 17 Apr 2023 Revised: 29 May 2023 Accepted: 6 Jun 2023 Published: 28 Jun 2023

Academic Editor: Simon X. Yang, Hongtian Chen Copy Editor: Yanbing Bai Production Editor: Yanbing Bai

Abstract
We investigate the cooperatability of the first-order leader-following multi-agent systems consisting of a leader and
a follower with multiplicative noises under Markov switching topologies. Each agent exhibits first-order linear dy-
namics, and there are multiplicative noises along with information exchange among the agents. What is more, the
communication topologies are Markov switching topologies. By utilizing the stability theory of the stochastic differ-
ential equations with Markovian switching and the Markov chain theory, we establish the necessary and sufficient
conditions for the cooperatability of the leader-following multi-agent systems. The conditions are outlined below: (i)
The product of the system parameter and the square of multiplicative noise intensities should be less than 1/2; (ii)
The transition rate from the unconnected graph to the connected graph should be twice the system parameter; (iii)
The transition rate from the connected graph to the unconnected graph should be less than a constant that is related
to the system parameter, the intensities of multiplicative noises, and the transition rate from the unconnected graph
to the connected graph. Finally, the effectiveness of our control strategy is demonstrated by the population growth
systems.

Keywords: Leader-following, Markov switching topologies, multiplicative noises, cooperatability

© The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0
International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, shar-
ing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you
give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate
if changes were made.

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