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Page 216 Li et al. Intell Robot 2023;3(2):213-21 I http://dx.doi.org/10.20517/ir.2023.13

formed bythe leader andthe follower, wherethe set ofnodes V = {0, 1} andthe set ofedges E( ( )) ⊆ V ×V.

0 0 ∈ R ,
2×2
Denote the neighbors of the th agent by N ( ( )). The adjacency matrix A( ( )) =
10 ( ( )) 0
where if 0 ∈ N 1 ( ( )), then 10 ( ( )) = 1, otherwise 10 ( ( )) = 0. The Laplacian matrix of G ( ( )) is given
by L( ( )) = D( ( )) − A( ( )), where D( ( )) = diag(0, 10 ( ( ))). Without losing generality, we assume

−
that the transition rate matrix of the Markov chain ( ) is the matrix = , where represents the
−
transition rate from the unconnected graph to the connected graph; represents the transition rate from the
connected graph to the unconnected graph.

3. ADMISSIBLE DISTRIBUTED COOPERATIVE CONTROL STRATEGY
In the real network, the relative state measurement information obtained by the follower from the leader is
often affected by noises. Therefore, for the leader−following multi-agent system (1)−(2), we assume that the
relative state measurement information has the following form

10 ( ) = 1 ( ) − 0 ( ) + 10 ( 1 ( ) − 0 ( )) 10 ( ), (3)

where 10 ( ) representsthemultiplicativemeasurementnoise, and 10 representstheintensityofmultiplicative
measurement noise.

We consider the following set of admissible distributed cooperative control strategies based on (3) and the
randomness of the communication topology

U = { = { ( ) = 10 ( ( )) 10 ( ), > 0} , ∈ R} . (4)

This paper primarily focuses on investigating the necessary and sufficient conditions for the cooperatability of
the first-order leader-following multi-agent systems. These systems are composed of a leader and a follower
and are subjected to multiplicative noises under Markov switching topologies.

The assumption and lemma required in this section are given below.

¯
Assumption1 The noise process 10 ( ) satisfies 0 10 ( )d = 10 ( ), ⩾ 0, where 10 ( ) is a one-dimensional
standard Brownian motion.

Lemma 1 [12] The solution of the Markov switching stochastic differential equations

d ( ) = ( ( )) ( )d + ( ( )) ( )d ( ) (5)

is mean square stable if and only if = diag( (1) ⊕ (1), . . . , ( ) ⊕ ( )) + diag( (1) ⊕ (1), . . . , ( ) ⊕
( )) + ⊗ 2 is a Hurwitz matrix, where = [ ] 1⩽ , ⩽ is the transition rate matrix of the Markov chain
T

( ). If ( ) = , we denote ( ( )) = ( ), ( ( )) = ( ), and = 1, . . . , .

4. MAIN RESULTS
Byleveragingthestabilitytheory ofstochasticdifferential equationswith MarkovianswitchingandtheMarkov
chain theory, we provide the necessary and sufficient conditions for the cooperatability of the leader-following
multi-agent systems.

Theorem 1 Suppose Assumption 1 is satisfied. In that case, there exists an admissible cooperative control
strategy denoted by ∈ U, which ensures that the follower can track the leader for any initial value under