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Page 214 Li et al. Intell Robot 2023;3(2):213-21 I http://dx.doi.org/10.20517/ir.2023.13

1. INTRODUCTION
In the past few decades, distributed cooperative control of multi-agent systems under fixed topologies has
attracted much attention from the system and control community [1–3] . However, in practical systems, the
communication networks connecting the agents often experience sudden interruptions and restoration. These
mutations lead to the changes in the structures or parameters of the system. Here, we describe this changing
topologybytheMarkovianswitchingtopology. Forsuchsystems, weusuallyusetheMarkovswitchingsystems
to describe them. In recent years, the stability of linear Markov switching systems has been widely studied [4–8] .
By Kronecker product and Lyapunov exponent, Mariton et al. [4] gave the necessary and sufficient conditions
for the moment stability and the almost sure stability of the system, respectively. Feng et al. [5] studied the
stochastic stability of the system and revealed the relationship between the moment stability and the almost
[6]
sure stability. Feng et al. studied the stabilization problem. The literature [7,8] investigated the robust stability
problems and gave sufficient and necessary conditions in the form of linear matrix inequalities for the mean
square stability.

[9]
In many real-world systems, it is inevitable for systems to be subjected to random noises . These noises
may change the trajectory of the system and even affect its stability. Therefore, an increasing number of re-
searchers have focused on studying the stability of the Markov switching stochastic systems. The stability of
linear Markov switching systems with stochastic noises was studied in previous literature [10–12] . Fragoso et
al. [10] studied the Markov switching systems with additive noises and provided the necessary and sufficient
conditions for the mean square stability of the system. On the other hand, the literature [11,12] explored the
Markov switching systems with multiplicative noises. By employing the operator theory, Dragan et al. [11]
derived the necessary and sufficient conditions in the form of linear matrix inequalities for the mean square
stability. Similarly, Sheng et al. [12] , also using the operator theory, presented a new necessary and sufficient
condition for the mean square stability. Using the Lyapunov method, Mao et al. [13] established a sufficient
condition for the ℎ moment exponential stability of the nonlinear Markov switching system and revealed
the relationship between the ℎ moment exponential stability and the almost sure exponential stability of the
system. In the context of nonlinear Markov switching systems, Deng et al. [14] addressed the problem of mean
square stabilization.

The stability theory of Markov switching systems with noises has numerous practical applications [15–17] . Previ-
ous studies [18–22] have focused on the distributed control problem of multi-agent systems with random noises
under Markov switching topologies. The literature [18,19] studied the distributed control problem of discrete-
timemulti-agentsystems. Bythestatespacedecompositionmethod,Huangetal. [18] gaveasufficientcondition
for almost sure consensus and mean square consensus, respectively. Zhang et al. [19] studied the mean square
consensus problem. The literature [20–22] considers the distributed control problem of continuous-time multi-
agent systems. Zhang et al. [20] studies the distributed control problem of multi-agent systems with first-order
integratordynamics. Lietal. [21] studiedthecontainmentcontrolproblem. Wangetal. [22] studiedmeansquare
consensus and almost sure consensus of higher-order multi-agent systems.

Compared with additive noises, multiplicative noises play a stabilizing role in the almost sure stability of sys-
tems [23] . Many scholars have studied the distributed control problem of multi-agent systems with multiplica-
tive noises [24–28] . However, as the state of the system is related to the Markov chain, we cannot write the expec-
tation of the product of the state variable and the indicative function in the form of the expected product. This
leads to the fact that the distributed control problem of multi-agent systems with multiplicative noises under
the Markov switching topology has not yet been solved. As a preliminary study, we study 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 has first-order linear dynamics, and there are multi-
plicative noises along with information exchange among agents. What is more, the communication topologies
are Markov switching topologies. Compared with existing literature [24–28] , we have revealed the influence