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Page 10 of 15 Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48

Remark 4 The difficulty of the proof is that the time delay is merely a Borel measurable function of rather
than a differentiable function of [13,28] . This means that the existing results [13,28] cannot be applied to SSS (1).
By selecting a suitable form of MLF, the existence and uniqueness of the global solution are initially proven via

an inequality scaling technique (i.e., Lemma 1). Subsequently, the -boundedness of the solution is obtained by
using the ADT method.
The following theorem demonstrates that a stronger result can be obtained under proper conditions.

Theorem 3 Let Assumptions 1-3 hold. If > 2 1 ∨ 2 2, then the solution of the controlled system (1) with the
initial value (2) is almost surely exponentially stable. That is,
1
lim sup ln(| ( )|) < 0 . . (29)
→∞
Proof. Let be any non-negative integer. Using the H ¥lder and Doob martingale inequalities [26] , we obtain

2 2
( sup | ( )| ) ≤ 4 | ( + 1)|
≤ ≤ +1
∫
+1
2
2
≤ 4[3 | ( )| + 3 | ( , ( ), ( ))|

∫
+1
2
+ 12 | ( , ( ), ( ))| ].

From condition (6), we have
∫
+1
2
2
( sup | ( )| ) ≤ 12 | ( )| + 4 (1 + | ( )| 2 1 + | ( − )| 2 1 )
≤ ≤ +1
∫
+1
+ 4 (1 + | ( )| 2 2 + | ( − )| 2 2 ) ,

where 4 is a positive constant. According to > 2 1 ∨ 2 2, we derive
2 1 2 1
| ( )| ≤ ( | ( )| ) ≤ 1 + | ( )| .
Similarly, we also have

2 2 ≤ 1 + | ( )| .
| ( )|
From (28), it follows that

∫ ∫ ∫
+1 +1 +1

| ( )| 2 1 ≤ 1 + | ( )| ≤ 1 + 3 − ˆ ≤ 5 − ˆ ,

where 5 is a positive constant, ˆ = − . Consequently, we can deduce that
J
( )

sup | ( )| 2 ≤ 5 − ˆ .
≤ ≤ +1
By the Doob martingale inequality, it follows that
∞ ( ) ∞
∑ ∑

sup | ( )| > −0.25 ˆ ≤ 5 −0.5 ˆ < ∞.
≤ ≤ +1
=0 =0