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Tan et al. Complex Eng Syst 2023;3:6 I http://dx.doi.org/10.20517/ces.2023.10 Page 9 of 23

In addition, the other scalars are given as follows

∑ T T
21
˜
Ξ = ( ) B P − 2(1 + )R − S 1 − S 2 − S 3 − S 4 ,

=1
˜
˜
Ξ 22 = −2(4 + )R − 2S 1 + 2S 2 − 2S 3 + 2S 4 + Ω , Ξ 31 = −S 1 − S 2 + S 3 + S 4 ,

˜
˜
Ξ 32 = −2(2 − )R + S 1 − S 2 − S 3 + S 4 , Ξ 33 = −Q − 4(2 − )R ,

˜
˜
˜
Ξ 41 = −S 3 − S 4 , Ξ ˜ 42 = S 3 − S 4 + 3(2 − )R , Ξ 43 = 3(2 − )R , Ξ 44 = −3(2 − )R ,

˜
˜
˜
Ξ 51 = 3(1 + )R , Ξ 52 = −S 2 + 3(1 + )R , Ξ 55 = −3(1 + )R ,

1
T
˜
˜
T
˜
T
T
˜
Ξ 61 = ( ) B P , Ξ 66 = − 0 1 Ω , Ξ +5,1 = ( ) B P , Ξ +5, +5 = − 0 Ω ,

[ ]
11 T
Γ = G P 0 0 0 0 0 · · · 0 ,
ˇ
−1
−1
−1
P = −( − 1) { P , · · · , P , ≠ , · · · , −1 P },
1 1 1
[ ∑ ]

1
21
Γ = R A B 0 0 0 B · · · B ,
=1
[ ]
31
Γ = R A 0 0 0 0 0 · · · 0 ,
[ ∑ ]

41
Γ = R 0 B 0 0 0 0 · · · 0 ,
=1
[ 1 ]
51
Γ = R 0 0 0 0 0 B · · · 0 ,
[ ]
61

Γ = R 0 0 0 0 0 0 · · · B ,
[ ]
71
Γ = R G 0 0 0 0 0 · · · 0 ,
ˇ
−1
−1
−1
R = −( − 1) {(1 + 2 + 3 + · · · + , +3 ) R 1 , · · · , (1 + 2 + 3 + · · · + , +3 ) R , ≠ ,
−1
· · · , (1 + 2 + 3 + · · · + , +3 ) R },
 0 S 1 + S 3 −S 1 + S 3 −S 3 0 0 · · · 0 
 
 
81  0 S 2 + S 4 −S 2 + S 4 −S 4 0 0 · · · 0
Γ =   ,
S 1 + S 3 −S 1 + S 3 0 0 −S 3 0 · · · 0
 
 S 2 + S 4 −S 2 + S 4 0 0 −S 4 0 · · · 0
 
−1
−1
−1
−1
ˆ
R = {−(1 − ) R , −3(1 − ) R , − R , −3 R }.

Proof: Define the following Lyapunov-Krasovskii functional ( ( ), ):

∑
( ( ), ) = [ 1 ( ( ), ) + 2 ( ( ), ) + 3 ( ( ), ) + 4 ( ( ), )], (20)
=1
where
T
1 ( ( ), ) = ( )P ( ) ( ),

∫ ∫ 0 ∫
T
T
2 ( ( ), ) = ( )Q ( ) ( ) + ( )Q ( ) ,

− − +
∫ 0 ∫
T

3 ( ( ), ) = ¤ ( )R ¤ ( ) ,

− +
1 2
4 ( ( ), ) = ( ).

2
According to Definition 1, we can get