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Page 4 of 15 Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48

Assumption 1 The time-varying delay is a Borel measurable function of from R + → [ 1 , ] with the prop-
erty that

( )
( ,Δ )
¯ = lim sup sup < ∞, (3)
≥− Δ
Δ→0 +
where 1 and are positive constants, ,4 = { ∈ R + : − ∈ [ , + 4)} and (·) denotes the Lebesgue
measure on R +.

Remark 1 Assumption 1 reveals that the time delay in SSS (1) is merely a Borel measurable function of time
, which means that it may be non-differentiable with respect to time . In most reported studies on SSSs (see,
e.g., [21–25] ), the time delay is always assumed to be a differentiable function and its time derivative should
¤
satisfy ≤ < 1 with being a positive constant. However, this condition is too conservative for practical
¯
¯
¤
application. Many time-delay functions in actual systems do not satisfy this assumption. For example, a time-
varying delay is defined as = 0.5 + 0.25| sin(10 )|. If is a Lipschitz continuous function with a Lipschitz
coefficient 2 ∈ (0, 1), namely, | − | ≤ 2 | − |, then for all 0 ≤ < < ∞. Then, satisfies Assumption 1
with ¯ = (1 − 2 ) . In particular, if is differentiable and its derivative is bounded by 2 ∈ (0, 1), then still
−1
satisfies Assumption 1. From a theoretical perspective, a large class of functions can satisfy Assumption 1. Note
that the constant ¯ must not be less than 1 (i.e., ¯ ≥ 1). This point can be obtained from the following lemma,
with = 1.
The following lemma provides a useful inequality to obtain the stability of the SSS (1) with time-varying delays,
and its proof can be found in [16] .

Lemma 1 [16] Let > 0 and : [ 0 − , − 1 ] → R be a continuous function. If Assumption 1 holds, then
+
∫ ∫ − 1
( − ) ≤ ¯ ( ) . (4)
0 −
0
The conditions for the existence and uniqueness of global solution are the local Lipschitz condition (LLC) and
the LGC (see, e.g., [4,7,20,26] ). In this paper, the highly nonlinear SSS (1) generally does not require the LGC.
Consequently, we must impose the PGC on it.
Assumption 2 (LLC & PGC) For any real number > 0, ∈ Γ, there exists a constant , > 0 such that

| ( , , ) − ( , ¯, ¯ )| ∨ | ( , , ) − ( , ¯. ¯ )| ≤ , (| − ¯| + | − ¯ |), (5)
for all , ¯, , ¯ ∈ R , where | | ∨ | ¯| ∨ | | ∨ | ¯ | ≤ . Moreover, there exist constants > 0, 1 > 1, 2 ≥ 1 such

that
1 + | | ),
1
| ( , , )| ≤ (1 + | |
2 2 (6)
| ( , , )| ≤ (1 + | | + | | ),

where ( , , ) ∈ R × R × R and ∈ Γ.
+
1,2
Assumption 3 Assume that there are two functions ∈ V ([− , ∞) × R × Γ; R + ) and ∈ ([− , ∞) ×
R ; R + ), as well as positive numbers 1 , 2 , 1 , 3 and real numbers 2 , 4, satisfying 1 > 2 , 3 > 4 and

> 2, > 1, such that for any ( , , , ) ∈ R + × R × R × Γ,

1 | | ≤ Λ( , , ) ≤ 2 | | , (7)

( , , ) ≤ ( , , ), ∀( , , ) ∈ R + × R × Γ, (8)
1
L ( , , , ) = ( , , ) + ( , , ) ( , , ) + trace{ ( , , ) ( , , ) ( , , )}
2

≤ − 1 ( ) + 2 ( ) − 3 | | + 4 | | , (9)