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Page 12 of 23 Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10

where

∑ ∑
T T
(2 ( )A R G ( ))

=1 =1, ≠

∑ ∑
T T −1 T T
≤ (( − 1) 2 ( )G R G ( ) + ( )A R A ( )),

2
=1 =1, ≠

∑ ∑ ∑
T T T
(2 ( − ( )) B R G ( ))

=1 =1 =1, ≠

∑ ∑
T T
≤ (( − 1) 3 ( − ( ))G R G ( − ( ))

=1 =1, ≠

∑ ∑
−1 T T T
+ ( − ( )) B R B ( − ( )),
3
=1 =1

∑ ∑ T ∑
T
T
( )
(2 ( ( )) B R G ( ))

=1 =1 =1, ≠

∑ ∑ ∑ ∑
T T −1 ( ) T T T ( )
≤ (( − 1) 4 ( )G R G ( ) + 4 ( ( )) B R B ( )),

=1 =1, ≠ =1 =1

∑ ∑ ∑ ∑ ∑
T T T T
( ( )G R G ( )) ≤ ( − 1) ( )G R G ( ).

=1 =1, ≠ =1, ≠ =1 =1, ≠
[ ] T
T ( ) T ( ) T , then we obtain
Let ( ) = ( ) ( ( )) · · · ( ( ))
1

∑
21 T
11 T
−1 41
41 T
−1 11
−1
−1
31 T
−1 21
−1 31
T
¤
( ( ), ) ≤ ( )[Γ 0 + (Γ ) P Γ + (Γ ) R Γ + (Γ ) R Γ + (Γ ) R Γ

3
2
=1
−1 51 T −1 51 −1 61 T ˜ −1 61 71 T ˇ −1 71 81 T ˆ −1 81
+ (Γ ) R Γ + (Γ ) R Γ + (Γ ) R Γ (Γ ) R Γ ] ( )
4 , +1

∑
T
= ( )Π ( ),

=1
ˇ
˜
ˆ
where Γ 0 = Ξ ( 1), Γ 0 = Ξ ( 2), Γ 0 = Ξ ( 3).

∑
Therefore, if Π < 0 and −Q + (ℎ)Q < 0, the interconnected semi-Markovian control system (10)
=1
with partially accessible TRs and dynamic METM is stochastically stable. Considering the partially accessible
transition rates, we will get the corresponding conclusion from the following three cases.
Case 1. If ∧ , ≠ ∅ and ∧ , ≠ ∅ , ∈ ∧ , , denote = ∑ ∈ ∧ , (ℎ) . Since ∧ , ≠ ∅ , then one
derives that < 0 , thus ∑ (ℎ)P can be presented as
=1

∑ ∑ ∑ ∑ ∑
(ℎ)
(ℎ)P = ( + ) (ℎ)P = (ℎ)P − P . (25)
∧ ∧ ∧ ∧ −
=1 ∈ , ∈ , ∈ , ∈ ,
(ℎ) ∧ ∑ (ℎ) ∧
Obviously, there exists 0 ≤ ≤ 1( ∈ , ) and ∧ = 1. So for ∀ ∈ , , there is
− ∈ , −

∑ ∑ (ℎ) ∑
(ℎ)P = ( (ℎ)(P − P )). (26)
∧ − ∧
=1 ∈ , ∈ ,