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Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10 Page 13 of 23

According to Schur complement, it is inferred that (14) is equivalent to Π < 0 and (15) is equivalent to
∑
−Q + (ℎ)Q < 0 .
=1

Case 2. If ∧ , ≠ ∅ and ∧ , ≠ ∅, ∈ ∧ , , denote = ∑ ∈ ∧ , (ℎ). Since ∧ , ≠ ∅, we know that
> 0, thus ∑ (ℎ)P can be presented as
=1

∑ ∑ ∑
(ℎ)P = (ℎ)P + (ℎ)P + (ℎ)P
∧ ∧
=1 ∈ , ∈ , , ≠ (27)
∑ ∑
(ℎ)
= (ℎ)P + (ℎ)P − ( (ℎ) + ) P .
∧ ∧ − (ℎ) −
∈ , ∈ , , ≠
∧ ∑ ∧
(ℎ) ) and (ℎ) , ≠ , there
It is obvious that 0 ≤ ≤ 1( ∈ , ∧ = 1. So for ∀ ∈ ,
− (ℎ)− ∈ , − (ℎ)−
is

∑ ∑ ∑
(ℎ)
(ℎ)P = ( (ℎ)(P − P ) + (ℎ)(P − P )). (28)
∧ − (ℎ) − ∧
=1 ∈ , , ≠ ∈ ,
∑
By applying Schur complement and (ℎ) = − =1, ≠ (ℎ), it is deduced that (16), (17) are equivalent
∑
to Π < 0 and (18) is equivalent to −Q + (ℎ)Q < 0 .
=1
Case 3. If ∧ , = ∅ and ∧ , ≠ ∅, ∈ ∧ , , assume there exists ≠ and ∈ ∧ , . Denote =
(ℎ) = (ℎ). Noting that < 0, ∑ (ℎ)P can be presented as
=1

∑ ∑ ∑
(ℎ)
(ℎ)P = (ℎ)P + (ℎ)P = (ℎ)P − P . (29)
∧ ∧ −
=1 ∈ , , ≠ ∈ , , ≠
It is obvious that ∑ ∧ (ℎ) = − (ℎ) = − > 0. So for ∀ ∈ ∧ , ≠ , there is
∈ , , ≠ ,

∑ ∑ (ℎ)
(ℎ)P = [ (ℎ)(P − P )] = (ℎ)(P − P ) = (ℎ)(P − P ). (30)
∧ −
=1 ∈ , , ≠
∑
By applying Schur complement and (ℎ) = − =1, ≠ (ℎ) , we konw that (19) is equivalent to Π < 0.

In summary, if inequalities (14) − (19) hold, the interconnected semi-Markovian control system (10) with
partially accessible TRs and dynamic METM is stochastically stable.

Remark 3. Theorem 1 designed sufficient conditions to ensure (10) is stochastically stable. However, it is dif-
ficult to directly use this result to acquire the controller gain matrices. Theorem 2 gives a LMI-based sufficient
criterion for the solvability.

Theorem 2. For given a positive real number > 0, > 0, > 0, 0 > 0, > 0 ( = 1, 2, 3, · · · , + 3),
∈ (0, 1) and (ℎ) ∈ [ , ], the system (10) with partially accessible TRs and dynamic METM is

stochastically stable if there are positive symmetric matrices > 0, > 0, Q > 0, Q > 0,R > 0, Ω > 0,

and matrices , S 1 , S 2 , S 3 and S 4 with proper dimensions, such that the linear matrix inequalities hold: