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Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48 Page 3 of 15

Figure 1. Framework of the paper.

surely exponential stability are presented for highly nonlinear SSSs with time-varying delays.

The remainder of this paper is organized as follows. An introduction of the model and important assumptions
aregiveninSection2. TheexistenceofauniqueglobalsolutionandstabilityanalysisarepresentedinSections3.
In Section 4, a simulation example is presented to validate our theoretical results. Finally, Section 5 concludes
the paper.

Note: In this paper, R + = (0, ∞), N + = 1, 2, · · · , , · · ·, N = N + ∪ {0} with being a positive finite integer,

R denotes the -dimensional real space. For ∈ R , | | = ( ∑ =1 2 ) 1 2 denotes the Euclidean norm of vector.
√

For any matrix ∈ R × , | | = denotes the trace norm of matrix , where is the transpose of
matrix and tr{ } denotes its trace. For > 0, ([− , 0]; R ) denotes the space of all continuous functions

from [− , 0] → R with the norm || || = sup − ≤ ≤0 | ( )|, C ([− , 0]; R ) denotes the family of all F 0-
F 0
measurable bounded C([− , 0]; R )-valued random variable = { ( ) : − ≤ ≤ 0}. Let ( , F , P) be a

complete probability space with a filtration {F } ≥0. ( )=( 1 ( ), · · ·, ( )) denotes an -dimensional F -
1,2
adapted Brownian motion, which is defined on a complete probability space. In addition, V denotes the

family of all non-negative functions ( , , ) : [− , ∞) × R × Γ → R +, which are first-order continuously

differentiable in and second-order continuously differentiable in . Let ([− , ∞) × R ; R + ) be the family of

continuous functions : [− , ∞)×R → R +. For real numbers and , ∧ = min{ , }, ∨ = max{ , }.
2. PRELIMINARIES
Model descriptions and assumptions are introduced in this section. In this study, we analyzed the following
highly nonlinear SSS with time-varying delays:
( ) = ( ) ( , ( ), ( − )) + ( ) ( , ( ), ( − )) ( ), (1)
with the initial value:

{ ( ) : − ≤ ≤ 0} = ∈ C ([− , 0]; R ), (2)
F 0
where > 0 is a constant and switching signal ( ) : [0, ∞) → Γ = {1, 2, · · · , } is a piecewise constant
function that is continuous from the right. In particular, it is a non-random function of . For ∈ [ , +1 ),
( ) = ∈ Γ, where is the th switching time instant and ∈ N. For each ∈ Γ, the mappings

: R × R × R → R and : R × R × R → R × are Borel-measurable functions. Compared with [13] ,
+
+
one of the merits of this paper is that the time delay is merely a Borel measurable function of and may be
non-differentiable. Precisely, we need to impose some requirements on the time-varying delay .