Appendix A. Proof of Proposition 1
Let us denote the Lyapunov (energy) function of perturbative gene regulatory network in equation (4) as
 
 T
[ ] X [ ] [ ] , for some symmetric positive definite matrix P

P T 
0 .



[

1]


V X t

X
T
[ t

1] [
  T
X [ ] [ ]


X
T t A (1
 
) 
T

X
T t

A (1
 
) 
T

   T
(1 ) [ ] X [ ] [ ]
(1
 
)
 
P

[ ]

0

(1
 
) 
T
 (1
 
) P 0
Based on quadratic stability [29], if the LMI in equation (5) holds, then the perturbative network is robustly stable.
Appendix B. Proof of Proposition 2
Let us denote the Lyapunov function of gene regulatory network under external stimulation signals in equation (7) as
 
 T
[ ] X [ ] [ ] , for some symmetric positive definite matrix P

P T 
0 . We have

[

1]


X
T
[ t

1] [

1]


ˆ
[ ]

[ ]
 
ˆ
[ ]

[ ]


X
T
[ ]
ˆ
T
ˆ
[ ]

X
T
[ ]
ˆ
T
[ ]
 T
[ ]
ˆ
[ ]
 T
[ ] [ ]


T
T
[ ]
ˆ
T
ˆ
[ ]
 T [ ]
ˆ
T [ ]

[ ]
ˆ
[ ]
 T [ ] [ ]
 
2 T
U t U t
 T
Y t Y t
 
2 T
[ ] [ ]

   


T
[ ]
ˆ
T
ˆ
[ ]
 T
[ ]
ˆ
T
[ ]

[ ]
ˆ
[ ]
 T
[ ] [ ]
T T
[ ]
  2 T
[ ] [ ]
 T
[ ] [ ]

 
 T
[ ] [ ]
 
2 T
[ ] [ ]
1

If the following inequality holds
X
T
[ ]
ˆ
T
ˆ
[ ]

X
T
[ ]
ˆ
T
[ ]

U
T
[ ]
ˆ
[ ]

U
T
[ ] [ ]
 T
X [ ]
T
[ ]
 
2
U
T
[ ] [ ]

X
T
[ ] [ ]

0 then, we get

[

1]
  
 T
[ ] Y [ ] [ ]
  2
U
T
[ ] [ ] inequality holds:
X
T
[ ]
ˆ
T
ˆ
[ ]

X
T
[ ]
ˆ
T
[ ]

U
T
[ ]
ˆ
[ ]

U
T
[ ] [ ]
 T
X [ ]
T
[ ]
 
2
U
T
[ ] [ ]

X
T
[ ] [ ]

0
The above inequality can be equivalent to the following inequality
X
T 

ˆ
T
ˆ  T 
P
ˆ
T
P
 
2
I

 
0
(A1)
So we sum the above inequality from 0 to ∞ to get
  
[0]

 t



0
T
Y [ ] [ ]
 
2 t



0
T
U [ ] [ ]

 t



0

Y
T

[ ] [ ] t



0
Y
T
[ ] [ ]


[0]

 
2 t



0
T
U [ ] [ ]

  t



0
T  T
Y [ ] [ ] X [0] PX [0]
  2 t



0
T
U [ ] [ ]
(A2) which is the inequality of response ability in equation (9) in the case X [0]

0
 t



0
T  
2 t



0
T

 t


0 t


0
T
T
 
2
The above results hold only when the inequality (A1) holds, i.e., the following
(A3)
The inequality (A3) holds for all [ ] and [ ] if and only if the following LMI holds.


ˆ
T
ˆ  T 
P
ˆ
T
 
2
P I

 
0
2