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Li et al. Intell Robot 2023;3(2):213-21 I http://dx.doi.org/10.20517/ir.2023.13 Page 215

of multiplicative noises and switching rates on the cooperatability of the system. To analyze this influence,
we delve into the stability theory of Markov switching systems with noises. Therefore, we introduced a new
lemma to address this issue. We establish the necessary and sufficient conditions for the cooperatability of the
leader-following multi-agent systems by combining the stability theory of the stochastic differential equation
with Markovian switching and the Markov chain theory. These conditions are outlined below: (i) The prod-
uct 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 value of the system
parameter; (iii) The transition rate from the connected graph to the unconnected graph should be lower than
a constant, which is related to the system parameter, the intensity of multiplicative noises, and the transition
rate from the unconnected graph to the connected graph.

The remaining sections of this paper are structured as follows: Section 2 formulates the problem. Section 3
presents the admissible cooperative distributed control strategy. Section 4 provides the main result. Section 5
includes a numerical simulation to demonstrate the effectiveness of our control laws. Section 6 concludes the
paper.

Notation: The symbols R and R + denote real and non-negative numbers, respectively. denotes the ×
dimensional identity matrix. The symbol diag{ 1 , . . . , } represents the block diagonal matrix with entries
being 1 , . . . , . Foragivenvectorormatrix , denotesitstranspose, and | | representsthedeterminant
T
of . Fortwomatrices and , ⊗ denotestheirKroneckerproduct,and ⊕ = ⊗ + ⊗ representsthe
that satisfies
Kronecker sum. Let (Ω, F , {F } ⩾ 0 , P) be a complete probability space with a filtration {F } ⩾ 0
the usual conditions, namely, it is right continuous and increasing while F 0 contains all P-null sets; ( ) =

( 1 ( ), . . . , ( )) denotes a -dimensional standard Brownian motion defined in Ω, F , {F } , P . For
⩾ 0
a given random variable , the mathematical expectation of is denoted by E[ ].

2. PROBLEM FORMULATIONS
Consider a leader-following multi-agent system consisting of a leader and a follower, where the leader and the
follower are indexed by 0 and 1, respectively. The dynamics of the leader is given by

¤ 0 ( ) = 0 ( ), (1)
where 0 ( ) ∈ R is the state, and ∈ R is a known constant.
+

The dynamics of the follower is given by

¤ 1 ( ) = 1 ( ) + ( ), (2)

where 1 ( ) ∈ R is the state, ( ) ∈ R is the input, and ∈ R and ∈ R/0 are known constants.
+
In this section, we assume that the topology graph is a Markovian switching topology. Let the switching signal
, P). The signal ( ) is a right continuous homogeneous
( ) be defined in the probability space (Ω, F , {F } ⩾ 0
Markov chain and has a finite state space S = {1, 2}. The matrix = [ ] 1⩽ , ⩽2 is the transfer rate matrix of
the Markov chain ( ) and satisfies
(
4 + (4), ≠ ,
( ( + 4) = | ( ) = ) =
1 + 4 + (4), = ,

where if ≠ , is the transition rate of the Markov chain from the state to the state with ⩾ 0; if = ,
2
Í (4)
= − ; 4 > 0 and lim = 0. We use G ( ( )) = (V, E( ( )), A( ( ))) to represent a weighted graph
≠ →∞ 4