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for any , ∈ O, ∈ N , then the system (18) is leader-following consensus in mean square sense under
[1, ]
−1
¯
−1
the controller gains = ( ) ( ) , where P , Ξ, Z , and Υ are defined in Theorem 1 and 2.

(0)
( )

By employing the same approach as Theorem 2, the Corollary 2 can be proved directly, omitting the proof
process here.
Remark 3 Theorem 2 realizes the consistent controller design of systems (6) and (8) under the channel fading
model (3). A set of sufficient conditions to ensure the consensus of systems and the existence of controller
gains is established based on the linear matrix inequality form. To solve the controller gain matrix , some
unknown variables are introduced into the inequality conditions of Theorem 2. The computational complexity
of solving the inequality conditions in Theorem 2 can be analyzed according to the total number of unknown
variables. It can be obtained by calculation that the total number of unknown variables in Theorem 2 is =

2
2
2
Σ [( +1) ]+Σ [ +2( ) ]. Itcanbefoundthatwiththeincreaseofthenumber oftopological
=1 =1

modes, the upper bound of the sojourn time of mode , and the number of agents , the computational

complexity of solving Theorem 2 also increases accordingly. Assuming that the upper bound , ∈ O
of the sojourn time is the same in each topological mode, the total number of unknown variables is =

( +1) + [ +2( ) ]. Similarly, thecomputationalcomplexityofsolvingtheinequalityconditions
2
2
2

in Theorem 3 can also be analyzed.
3.3. Extension results
In this subsection, it is assumed that the information of the semi-Markov kernel Π( ) is not completely acces-
sible. By using the similar method in [40] , the index set O of the semi-Markov chain ( ) can be partitioned
into the following form:
O = { ∈ O|if ( ) is accessible },
O = { ∈ O|if ( ) is inaccessible },
O = { ∈ O|if ( ) is accessible, is inaccessible },
O = { ∈ O|if ( ) is inaccessible, is accessible },
O = { ∈ O|if ( ) is inaccessible, is inaccessible },
where O = O ∪ O , O = O ∪ O ∪ O , O ∩ O = ∅, O ∩ O = ∅.
In this paper, only the case of O = O ∪ O for incompletely accessible semi-Markov kernel is considered. In
otherwords,thetransitionprobability ofEMCandtheprobabilitydensityfunction ( ) ofsojourn-time
are partially accessible. Before presenting the results of this subsection, we make the following assumptions,
which are crucial for subsequent derivations.

Assumption 3 Given a positive scalar , the selection of the upper bounds for sojourn time can be guar-
anteed by the following prerequisite:

Σ ( ) ≥ , 0 < < 1, ∀ ∈ O, ∀ ∈ O ∪ O (24)
=1
Then, the following Theorem proposes the leader-following mean-square consensus conditions for systems (6)
and (8) under incompletely accessible semi-Markov kernel of switching topologies.

( )
˜
Theorem 3 Given a scalar ∈ N ≥1, if there exist a scalar ˆ > 0 and sets of symmetric matrices ∈
˜
˜
R × ≻ 0, ( ) ∈ R × ≻ 0, and matrices ∈ R × ˜ × , , ∈ O, ∈ N such that the
, ∈ R
[0, ]
following inequalities
 ˜ ( −1) ) ∗ ∗ ∗ 
 −( ⊗ 
 0 2 ∗ 
˜  − ˆ ∗  (25)
Ψ =  ˇ ˜ −( ) ˜ ˜  ≺ 0
 Ξ Ψ 23 −P − Z − Z ∗ 
 
 ( ⊗ ) 0 0 − 
 