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Page 218 Li et al. Intell Robot 2023;3(2):213-21 I http://dx.doi.org/10.20517/ir.2023.13

By > 2 and (8), we get

2
2 2 2 + 2 < − 2 . (9)
10
2 −
By Condition (C ), we obtain
2
2
2 2
+ 2 < + − 4 . (10)
10
By ⩾ 0, > 0 and > 2 , we get
2 2
+ − 4 − ( − 2 ) = − 2 + + > 0, (11)
2 − − 2
2
which implies + − 4 > 2 − − 2 .

By (9), (10), and (11), we have

2
2 2
2
+ 2 < − 2 . (12)
10 2 −
2 2 2
Denote ( ) = + 2 − + 2 and = . By (12), we know that ( ) < 0 has a solution for variable
10 2 −
2
2 2 ( −2 )(1−2 )
. By ( ) < 0, we have Δ = 4 − 4 (− + 2 ) > 0. By Δ > 0, we get < 10 . Combining
10 2 − 2 2
10
2
0 ⩽ < ( −2 )(1−2 ) and > 2 , we have 2 2 < 1. In summary, we obtain 2 2 < 1, > 2 ,
10
2 2 10 10
10
2
( −2 )(1−2 )
0 ⩽ < 10 .
2 2
10
2
( −2 )(1−2 )
Sufficiency: By 2 2 < 1, > 2 and 0 ⩽ < 10 , we get (12). By (12), we have ∈
10 2 2
√ √ 10
−2( −2 )− −2( −2 )+ , where = 4( − 2 ) − 4( − 2 ) [2 ( − 2 ) + 2 ]. From the value range of
,
2
2
2( −2 ) 2 10 2( −2 ) 10 10
2
, it can be seen that Condition (C ) and Condition (C ) hold. Therefore, since the real parts of the zero
2
1
point of ( ) are less than −2 , it can be concluded that all eigenvalues of have negative real parts. Lemma
1 implies that the system (6) is mean square stable. Therefore, there exists an admissible cooperative control
strategy ∈ U, such that for any initial value, the follower can track the leader under the distributed control
law .
2
( −2 )(1−2 )
Remark 1 The conditions > 2 , 2 2 < 1 and 0 ⩽ < 10 stated in Theorem 1 highlight the
10 2 2
10
influenceofmultiplicativenoisesandboththetransitionrates and onthecooperatabilityofthesystem. Itis
shown that smaller multiplicative noises, lower transition rate , and higher transition rate are all favorable
for the cooperatability of the system. Moreover, the transition rates and have lower and upper bounds,
respectively. What is more, the noises and the system parameters satisfy the corresponding inequality.
We have the following corollary for the case without measurement noises.
Corollary 1 Suppose Assumption 1 and 10 = 0 hold. In that case, there exists an admissible cooperative
control strategy denoted by ∈ U, such that for any initial value, the follower can track the leader under the
distributed control law , if and only if > 2 .
5. NUMERICAL SIMULATION
In this section, we will use a numerical example to demonstrate the effectiveness of our control laws.