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

Lemma 1( [38] ): For a given scalar ∈ (0, 1), the continuous function ( ) ∈ (0, ] and ¤ ( ) : [− , 0) →

R , there exist positive symmetric matrices R ∈ R × and S ∈ R 2 ×2 to make the inequality hold

∫
T ˜
T
T ˜
T

¤ ( )R ¤ ( ) ≥ R 1 1 + R 2 2 + 2 S 2 , (12)

1
1
2
−
where
˜
˜ −1
˜
˜
˜
˜
˜
T ˜ −1
T
˜
R 1 = R + (1 − )(R − S R S ), R 2 = R + (R − S R S ), R = {R , 3R },

[ ] [ ]
( − ( )) − ( − ) ( ) − ( − ( )
1 = , 2 = ,
( − ( )) + ( − ) − 2 1 ( ) + ( − ( )) − 2 2
[ ] ∫ ∫ − ( )
S 1 S 2 1 1
S = , 1 = ( ) , 2 = ( ) .
S 3 S 4 ( ) − ( ) − ( ) −
Lemma 2( [39,40] ): For a real scalar > 0, the matrices > 0, > 0, the following inequality holds
2
− −1 ≤ −2 + , ∈ . (13)

3. MAIN RESULTS
Our purpose is to co-design the memory controller (9) and dynamic METM (4) so that system (10) with
partially accessible transition rates is stochastically stable. By utilizing the Lyapunov function method, some
sufficient conditions that insure the stochastic stability of the interconnected semi-Markovian system (10) are
given. Then, a controller design scheme based on LMI is given in Theorem 2.
Theorem 1. For given positive real number > 0, > 0, > 0, 0 > 0, > 0 ( = 1, 2, 3, · · · , + 3),
and ∈ (0, 1), the interconnected semi-Markovian jump control system (10) is said to be randomly stable
with partially accessible transition rates and dynamic METM if there are positive symmetric matrices P > 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

following matrix inequalities hold:
Case 1. If ∧ , ≠ ∅ and ∧ , ≠ ∅, ∈ ∧ , , for ∀ ∈ ∧ , , we have
 ˜ 
Ξ
 ∗ ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ 
 11 ˇ 
Γ −P ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ 
 21 
Γ 0 −R ∗ ∗ ∗ · · · ∗ ∗ ∗ 
 31 
Γ 0 0 − 2 R ∗ ∗ · · · ∗ ∗ ∗ 
 
 41 0 0 0 − 3 R ∗ · · · ∗ ∗ ∗ 
Γ
  (14)
 51 0 0 0 0 · · · ∗ ∗ ∗  < 0,
Γ
 − 4 R 
 . . . . . . . . . . 
 . . . . . . . . .
 . . . . . . . . . . 
. 
 61 
Γ 0 0 0 0 0 · · · − , +3 R ∗ ∗ 
 71 
ˇ
Γ 0 0 0 0 0 · · · 0 −R ∗ 
 
 Γ 81 0 0 0 0 0 · · · 0 0 −R
ˆ 
 
∑
−Q + (ℎ)(Q − Q ) < 0, (15)
∧
∈ ,