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

1. INTRODUCTION
Switched systems are important dynamic systems. The idea of switching has been widely applied in various
[2]
[3]
[1]
fields, such as aircraft attitude control , ecological dynamics , and financial markets . With the increasing
complexity of system architectures, dynamical analysis of switched systems has attracted significant academic
interest. A switched system consists of a family of continuous-time dynamics, discrete-time dynamics, and
switching rules between subsystems. According to the switching signal features, switched systems are divided
intotwocategories, namely, deterministicswitchedsystemsandrandomlyswitchedsystems. Manyresearchers
[4]
have focused on stabilization and stability analyses of various switched systems. For example, in , a series of
results on stochastic differential equations (SDEs) with Markovian switching was obtained. In particular, the
[5]
authors have provided some useful stability criteria. In , the authors studied the input-to-state stability of
time-varying switched systems by employing the ADT method coupled with the MLF approach. The authors
[6]
of investigated the stability of switched stochastic delay neural networks with all unstable subsystems based
[7]
on discretized Lyapunov-Krasovskii functions (DLKFs). In , a novel Lyapunov function was designed to
[8]
ensure a non-weighted L 2 gain for switched systems with asynchronous switching. In , a hidden Markov
model was proposed to study the finite region ∞ asynchronous control problem for two-dimensional Markov
jump systems. Other interesting researches on switched systems can be found in [9–11] and references therein.

The linear growth condition (LGC) is crucial for ensuring the existence of a global solution for a stochastic
system. However, many stochastic systems do not satisfy LGC. Hence, the solution of a stochastic system
may explode in a finite time. Recently, the stability of stochastic systems without LGC has drawn consider-
able attention. For instance, the authors of [12] investigated the stability and boundedness of nonlinear hybrid
stochastic differential delay equations without LGC based on a Lyapunov function approach. By introducing a
polynomial growth condition (PGC), [13] discussed the stabilization problem of highly nonlinear hybrid SDEs.
The input-to-state practically exponential stability in the sense of mean square was introduced in [14] . Suffi-
cient conditions for stability have been obtained. Additionally, other meaningful results were reported in [15]
and [16] .

Time-delayisanimportantfactorthataffectsdynamical performancesofstochastic systems. Byconstructinga
suitableLyapunovfunction, theauthorsof [12] studiedthestabilityandboundednessofhighlynonlinearhybrid
stochastic systems with a time delay. The authors of [17] used the ADT method to study the stability problem of
SSSs, where the switching signals are deterministic. Based on the stability criteria for stochastic time-delay sys-
tems, the authors of [18] introduced a suitable Lyapunov-Krasovskii (L-K) functional, and discussed the global
probabilistic asymptotic stability of the closed-loop system. In [19] , the Razumikhin approach was presented to
study the exponential stability of a class of impulsive stochastic delay differential systems. Using the piecewise
dynamic gain method, the authors of [20] studied the global uniform ultimate boundedness of switched linear
time-delay systems. Motivated by the aforementioned literature, the stability of highly nonlinear SSSs with
time-varying delays is studied in this paper. Figure 1 shows the framework of this paper.

The challenges of this article lie in the following two parts: (1) The time delay studied here is merely a Borel
measurable function of time . That is to say, it may be non-differentiable with respect to time , which means
that the existing methods regarding constant delays or differentiable delays are no longer applicable; (2) Rather
thanaMarkovianswitchingsignal,adeterministicswitchingsignalisinvolvedinthestudiedsystem,indicating
thatMarkovianswitchedsystemsbasedM-matrixmethodisinvalid. Toaddresstheinfluencesofdeterministic
switching signals, an ADT method coupled with the MLF approach is utilized in our stability analysis.

The main advantages of this paper are as follows:
(1) Without the LGC, the existence and uniqueness of a global solution is proven for highly nonlinear SSSs,
where a deterministic switching signal rather than a Markovian switching signal is considered.
(2) By integrating the ADT method and MLF approach, the th moment exponential stability and almost