Stability analysis for highly nonlinear switched stochastic systems with time-varying delays

1. INTRODUCTION

Switched systems are important dynamic systems. The idea of switching has been widely applied in various fields, such as aircraft attitude control ^[1], ecological dynamics ^[2], and financial markets ^[3]. 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 into two categories, {namely}, deterministic switched systems and randomly switched systems. Many researchers have focused on stabilization and stability analyses of various switched systems. For example, in ^[4], a series of results on stochastic differential equations (SDEs) with Markovian switching was obtained. In particular, the authors have provided some useful stability criteria. In ^[5], 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 of ^[6] investigated the stability of switched stochastic delay neural networks with all unstable subsystems based on discretized Lyapunov-Krasovskii functions (DLKFs). In ^[7], a novel Lyapunov function was designed to ensure a non-weighted $$ \mathcal{L}_2 $$ gain for switched systems with asynchronous switching. In^[8], a hidden Markov model was proposed to study the finite region $$ H_\infty $$ 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 considerable 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]. Sufficient conditions for stability have been obtained. Additionally, other meaningful results were reported in ^[15] and ^[16].

Time-delay is an important factor that affects dynamical performances of stochastic systems. By constructing a suitable Lyapunov function, the authors of ^[12] studied the stability and boundedness of highly nonlinear hybrid 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 systems, 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.

Figure 1. Framework of the 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 $$ t $$ . That is to say, it may be non-differentiable with respect to time $$ t $$ , which means that the existing methods regarding constant delays or differentiable delays are no longer applicable; (2) Rather than a Markovian switching signal, a deterministic switching signal is involved in the studied system, indicating that Markovian switched systems based M-matrix method is invalid. To address the influences of deterministic 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 $$ q $$ th moment exponential stability and almost 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 are given in Section 2. The existence of a unique global solution and stability analysis are presented in Sections 3. In Section 4, a simulation example is presented to validate our theoretical results. Finally, Section 5 concludes the paper.

Note: In this paper, $$ \mathbb{R}_{+}=(0, \infty) $$ , $$ \mathbb{N}_{+}=1, 2, \cdots, \kappa, \cdots $$ , $$ \mathbb{N}=\mathbb{N}_{+}\cup\{0\} $$ with $$ \kappa $$ being a positive finite integer, $$ \mathbb{R}^{n} $$ denotes the $$ n $$ -dimensional real space. For $$ x\in\mathbb{R}^{n} $$ , $$ |x|=\big(\sum_{i=1}^nx_i^2\big)^\frac{1}{2} $$ denotes the Euclidean norm of vector. For any matrix $$ A\in\mathbb{R}^{n\times n} $$ , $$ |A| =\sqrt{A^TA} $$ denotes the trace norm of matrix $$ A $$ , where $$ A^T $$ is the transpose of matrix $$ A $$ and $$ {\rm tr}\{A\} $$ denotes its trace. For $$ \tau>0 $$ , $$ C([-\tau, 0];\mathbb{R}^n) $$ denotes the space of all continuous functions $$ \varphi $$ from $$ [-\tau, 0]\to\mathbb{R}^n $$ with the norm $$ ||\varphi||=\sup_{-\tau\leq u\leq0}|\varphi(u)| $$ , $$ \mathcal{C}_{\mathcal{F}_0}^b([-\tau, 0];\mathbb{R}^n) $$ denotes the family of all $$ \mathcal{F}_0 $$ -measurable bounded $$ \mathcal{C}([-\tau, 0];\mathbb{R}^n) $$ -valued random variable $$ \xi=\{\xi(\theta):-\tau\leq\theta\leq0\} $$ . Let $$ (\varOmega, \mathcal{F}, \mathbb{P}) $$ be a complete probability space with a filtration $$ \{\mathcal{F}_{t}\}_{t\geq0} $$ . $$ B(t) $$ =($$ B_1(t) $$ , $$ \cdots $$ , $$ B_m(t) $$ ) denotes an $$ m $$ -dimensional $$ \mathcal{F}_t $$ -adapted Brownian motion, which is defined on a complete probability space. In addition, $$ \mathcal{V}^{1, 2} $$ denotes the family of all non-negative functions $$ V(t, x, i):[-\tau, \infty)\times\mathbb{R}^n\times\Gamma\to\mathbb{R}_+ $$ , which are first-order continuously differentiable in $$ t $$ and second-order continuously differentiable in $$ x $$ . Let $$ C([-\tau, \infty)\times\mathbb{R}^n;\mathbb{R}_+) $$ be the family of continuous functions $$ W:[-\tau, \infty)\times\mathbb{R}^n\to\mathbb{R}_+ $$ . For real numbers $$ a $$ and $$ b $$ , $$ a\wedge b=\min\{a, b\} $$ , $$ a\vee b=\max\{a, b\} $$ .

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:

with the initial value:

where $$ m>0 $$ is a constant and switching signal $$ \sigma(t):[0, \infty)\to \Gamma=\{1, 2, \cdots, \kappa\} $$ is a piecewise constant function that is continuous from the right. In particular, it is a non-random function of $$ t $$ . For $$ t\in[t_m, t_{m+1}) $$ , $$ \sigma(t)=i_m\in\Gamma $$ , where $$ t_m $$ is the $$ m $$ th switching time instant and $$ m\in\mathbb{N} $$ . For each $$ i\in\Gamma $$ , the mappings $$ f_i:\mathbb{R}^{+}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n $$ and $$ g_i:\mathbb{R}^{+}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^{n\times m} $$ are Borel-measurable functions. Compared with ^[13], one of the merits of this paper is that the time delay $$ \delta_t $$ is merely a Borel measurable function of $$ t $$ and may be non-differentiable. Precisely, we need to impose some requirements on the time-varying delay $$ \delta_t $$ .

Assumption 1. The time-varying delay $$ \delta_t $$ is a Borel measurable function of $$ t $$ from $$ \mathbb{R}_+\to[m_1, m] $$ with the property that

where $$ m_1 $$ and $$ m $$ are positive constants, $$ M_{s, \bigtriangleup}=\{t\in\mathbb{R}_+: t-\delta_t \in [s, s+\bigtriangleup)\} $$ and $$ \mu(\cdot) $$ denotes the Lebesgue measure on $$ \mathbb{R}_+ $$ .

Remark 1 Assumption 1 reveals that the time delay in SSS (1) is merely a Borel measurable function of time $$ t $$ , which means that it may be non-differentiable with respect to time $$ t $$ . In most reported studies on SSSs (see, e.g., ^[21-25]), the time delay $$ \delta_t $$ is always assumed to be a differentiable function and its time derivative $$ \dot{\delta}_t $$ should satisfy $$ \dot{\delta}_t\leq\bar{\delta}<1 $$ with $$ \bar{\delta} $$ 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 $$ \delta_t $$ is defined as $$ \delta_t=0.5+0.25|\sin(10t)| $$ . If $$ \delta_t $$ is a Lipschitz continuous function with a Lipschitz coefficient $$ m_2\in (0, 1) $$ , namely, $$ |\delta_t-\delta_s|\leq m_2|t-s| $$ , then for all $$ 0\leq s. Then, $$ \delta_t $$ satisfies Assumption 1 with $$ \bar{m}=(1-m_2)^{-1} $$ . In particular, if $$ \delta_t $$ is differentiable and its derivative is bounded by $$ m_2\in (0, 1) $$ , then $$ \delta_t $$ still satisfies Assumption 1. From a theoretical perspective, a large class of functions $$ \delta_t $$ can satisfy Assumption 1. Note that the constant $$ \bar{m} $$ must not be less than $$ 1 $$ (i.e., $$ \bar{m}\geq1 $$ ). This point can be obtained from the following lemma, with $$ \psi=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 $$ T>0 $$ and $$ \psi :[t_0-m, T-m_1]\to\mathbb{R}^+ $$ be a continuous function. If Assumption 1 holds, then

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 $$ b>0, i\in \Gamma $$ , there exists a constant $$ K_{b, i}>0 $$ such that

for all $$ x, \bar{x}, y, \bar{y}\in\mathbb{R}^n $$ , where $$ |x|\vee|\bar{x}|\vee|y|\vee|\bar{y}|\leq b $$ . Moreover, there exist constants $$ K>0, \alpha_1>1, \alpha_2\geq1 $$ such that

where $$ (t, x, y)\in\mathbb{R}^{+}\times\mathbb{R}^{n}\times\mathbb{R}^{n} $$ and $$ i\in \Gamma $$ .

Assumption 3 Assume that there are two functions $$ \varLambda\in\mathcal{V}^{1, 2}([-m, \infty)\times\mathbb{R}^n\times\Gamma;\mathbb{R}_+) $$ and $$ W\in C([-m, \infty)\times \mathbb{R}^n;\mathbb{R}_+) $$ , as well as positive numbers $$ a_1, a_2, \lambda_1, \lambda_3 $$ and real numbers $$ \lambda_2, \lambda_4 $$ , satisfying $$ \lambda_1>\lambda_2, \lambda_3>\lambda_4 $$ and $$ q>2, \mu_{i}>1 $$ , such that for any $$ (t, x, y, i)\in \mathbb{R}_{+}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\Gamma $$ ,

where $$ y=x(t-\delta_t) $$ and

Moreover, assume that there exists a constant $$ \varepsilon>0 $$ , such that \\

Remark 2 The system studied in this research has the property of high nonlinearity. In other words, the LGC is removed from the SSS (1), which makes the considered system more general. Without the LGC, the solution of a stochastic system may explode in a finite time. To ensure the existence of a global solution, a PGC (i.e., condition (6)) is imposed on the SSS (1) (see, e.g., ^{[13, 27, 28]}). Therefore, the system (1) we studied obeys the LLC (i.e., condition (5)) and the PGC. By combining the MLF approach and ADT method, we then prove the existence and uniqueness of the global solution.

Before presenting the main results, the definition of ADT is revisited.

Definition 1^[28] For a switching signal $$ \sigma(t) $$ and any $$ t\geq s\geq 0 $$ , $$ T_i(t, s) $$ and $$ N_i(t, s) $$ denote the whole running time and the switching number of the i-th subsystem over the interval $$ [s, t] $$ , respectively, $$ i\in\Gamma $$ . Then, the following inequality holds:

where $$ \mathcal{J}_{ai}>0 $$ is called the mode-dependent ADT and $$ N_{0i}>0 $$ is the mode-dependent chatter bound.

3. MAIN RESULTS

In this section, we prove the existence of a unique global solution for a highly nonlinear SSS (1) by using the ADT and MLF approaches. Then, both the $$ q $$ th moment exponential stability and almost surely exponential stability are provided for a highly nonlinear SSS (1).

Theorem 1 Under Assumptions 1-3, if there exists a constant $$ \varepsilon>0 $$ such that

Then, for any initial data (2), there exists a unique global solution $$ x(t) $$ for the SSS (1) on $$ [-m, \infty) $$ , and the solution satisfies

Proof. We divide the whole proof into two steps. In step 1, for all $$ i\in\mathbb{S} $$ , we prove that the $$ i $$ -th subsystem with the initial value $$ x_i(0) $$ has a unique global solution $$ x_i(t) $$ . In step 2, when each subsystem has a unique global solution, the SSS (1) with a deterministic switching signal has a unique global solution $$ x(t) $$ on $$ [-m, \infty) $$ .

Step 1. For all $$ i\in\mathbb{S} $$ , the control system becomes

where $$ y_i(t)=x_i(t-\delta_t) $$ . Under the LLC, system (14) has a unique maximal global solution on $$ [-m, \rho_{\infty}^i) $$ , denoted as $$ x_i(t) $$ , where $$ \rho_{\infty}^i $$ is the explosion time. Then, we prove $$ \rho_{\infty}^i=\infty $$ a.s. Thus, it is necessary to define the stopping time sequence. Let $$ k_0 $$ be a constant sufficiently large to satisfy $$ k_0>|x_i(0)| $$ . For any integer $$ k\geq k_ 0 $$ , we define the stopping time sequence as follows:

Clearly, $$ \delta_{k, i} $$ increases as $$ k\to\infty $$ and therefore we set $$ \delta_{\infty, i}:=\lim\limits_{k\to\infty}\delta_{k, i} $$ . Observe that $$ \delta_{\infty, i}\leq\rho_{\infty, i} $$ a.s. Thus, $$ \delta_{\infty, i}=\infty $$ , a.s., which yields $$ \rho_{\infty, i}=\infty $$ a.s. From the It$$ \hat{o} $$ formula and condition (9), it is easily proven that

By Lemma 1, we have

and

Hence,

where

is a finite constant. Applying (10) and (11) from Assumption 3, we can deduce that

Recalling the condition (7), we can get

This implies

We observe that

Letting $$ k\to\infty $$ yields that $$ P(\delta_{\infty, i}\leqslant t)=0 $$ . Hence, $$ \delta_{\infty, i}=\infty $$ a.s. Therefore, we have $$ \rho_{\infty, i}=\infty $$ a.s. This implies that the unique solution for the $$ i $$ -th subsystem (14) will not explode in finite time.

Step 2. This section proves the existence of a unique global solution for SSS (1). Let $$ k_0>0 $$ be a sufficiently large integer, such that $$ k_0>|x_i(0)| $$ , where $$ |x_i(0)| $$ is the initial data of the $$ i $$ -th subsystem. For any integer $$ k\geq k_0 $$ , we define the stopping time sequence as follows:

Clearly, $$ \delta_k^n $$ increases as $$ k\to\infty $$ . For $$ t\in[t_0, t_1) $$ , $$ \sigma(t)=i_0 $$ , using the It$$ \hat{o} $$ formula, we have

where $$ \varXi(s, i_0)=\varepsilon\Lambda(s, x(s), i_0)+\mathcal{L}\Lambda(s, x(s), y(s), i_0) $$ . Letting $$ t=t_1 $$ , according to condition (8) in Assumption 3, we derive that

For $$ t\in[t_1, t_2) $$ , $$ \sigma(t)=i_1 $$ , we obtain

Combining (18) and (19), it implies that

For $$ t\in[t_{m-1}, t_m) $$ and $$ \sigma(t)=i_{m-1} $$ , we assume that

By mathematical induction, for $$ t\in[t_m, t_{m+1}) $$ and $$ \sigma(t)=i_m $$ , we have

It follows from (8) and (21) that

Because $$ \mu_i>1 $$ , we obtain from (23) that

Similar to the proof stated in Part 1, we can derive

where

is finite. Then,

Recalling condition (7), we obtain

This implies

Letting $$ k\to\infty $$ , we observe that $$ P(\delta_{\infty}^m\leq t)=0 $$ and hence $$ \delta_{\infty}^m\geq t $$ a.s. We let $$ k\to\infty $$ in (24) to obtain

Using Definition 1, we have that for $$ t\geq 0 $$ and $$ i\in \Gamma $$ ,

where $$ C_2=C_1\mu_i^{N_{0i}} $$ . This implies

Therefore, for all $$ m\in \mathbb{N} $$ , we obtain

This means that the unique solution $$ x(t) $$ will not explode for $$ t\in[t_m, t_{m+1}) $$ and $$ m\in \mathbb{N} $$ . Hence, there exists a unique global solution $$ \{x(t), t\geq0\} $$ for SSS (1). Moreover, from (25), we obtain that

The proof is completed.

Remark 3 To deal with the time-varying delay $$ \delta_t $$ , some new inequalities (e.g., see (15) and (16) for details) are constructed in the proof for Theorem 3.1. Compared with the results reported in existing studies ^[21-25], the time delay $$ \delta_t $$ in this paper is merely a Borel-measurable function, which invalidates these existing methods. By virtue of Lemma 1, a more general form of time delay can be imposed on system (1).

We now refer to the equation (25) in the proof of Theorem 1. The following theorem provides sufficient conditions for the $$ q $$ th exponential stability of system (1).

Theorem 2 Under the same conditions as those considered in Theorem 1, the solution of system (1) with the initial value (2) is $$ q $$ th moment exponentially stable. That is,

Proof. Applying (25) yields

Recalling condition (7), we have

Hence, from (12), we observe that

where $$ C_3=\dfrac{C_2}{a_1} $$ , which is the required assertion in (27). The proof is completed.

Remark 4 The difficulty of the proof is that the time delay $$ \delta_t $$ is merely a Borel measurable function of $$ t $$ rather than a differentiable function of $$ t $$ ^{[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 $$ L^q $$ -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 $$ q>2\alpha_1\vee2\alpha_2 $$ , then the solution of the controlled system (1) with the initial value (2) is almost surely exponentially stable. That is,

Proof. Let $$ k $$ be any non-negative integer. Using the H$$ \ddot{o} $$ lder and Doob martingale inequalities ^[26], we obtain

From condition (6), we have

where $$ C_4 $$ is a positive constant. According to $$ q>2\alpha_1\vee2\alpha_2 $$ , we derive

Similarly, we also have

From (28), it follows that

where $$ C_5 $$ is a positive constant, $$ \hat{\varepsilon}=\varepsilon-\dfrac{ln\mu_{i}}{\mathcal{J}_{ai}} $$ . Consequently, we can deduce that

By the Doob martingale inequality, it follows that

From the well-known Borel-Cantelli lemma^[4], it follows that for almost all $$ \omega\in\Omega $$ , there exists a positive integer $$ k_0=k_0(\omega) $$ such that

Therefore, for almost all $$ \omega\in\Omega $$ ,

Then, we can obtain

which is the required assertion in (29). Thus, the proof is completed.

So far, we can conclude that under Assumptions 1-3, system (1) is not only $$ q $$ th moment exponentially stable but also almost surely exponentially stable.

Remark 5 In general, for a stochastic nonlinear system, the $$ q $$ th moment exponential stability does not imply almost surely exponential stability without any imposed conditions. However, this result can be ensured using the PGC (6). Similar arguments can be found in ^{[4, 13]}.

Remark 6 In this paper, the highly nonlinear SSSs with time-varying delays are considered, in which the switching signal is deterministic and differs from those considered in^{[13, 16, 29-32]}. In the current study on stochastic systems with Markovian switching ^{[13, 16, 29-32]}, $$ M $$ matrix theory is an efficient tool for achieving stochastic stability. However, this method is not valid for SSS (1) because a deterministic switching signal rather than the Markovian switching signal is involved in (1). In this paper, a new stability analysis based on the ADT method coupled with the MLF approach is developed for SSSs. In our proof, the Lyapunov functions do not need to be specified initially, which increases the flexibility for the choice of Lyapunov functions in practice.

4. NUMERICAL EXAMPLE

In this section, a numerical example is presented to validate the derived results. Consider the following highly nonlinear SSS with a time-varying delay:

where the time-varying delay $$ \delta_t=\frac{1}{2}+\frac{1}{4}|\sin(10t)| $$ , the initial data $$ x(\theta)=\xi=0.1\pi $$ with $$ -\frac{3}{4}\leq \theta\leq 0 $$ , and

In addition, we set $$ \Lambda(t, x, 1)=x^6 $$ and $$ \Lambda(t, x, 2)=\frac{11}{12}x^6 $$ . It is not difficult to verify that Assumption 1 holds with $$ m_1=\frac{1}{2} $$ , $$ m=\frac{3}{4} $$ , and $$ \bar{m}=\frac{4}{3} $$ , and $$ f_1, f_2, g_1, g_2 $$ satisfy Assumption 2. Then, we have $$ a_1=\frac{11}{12} $$ , $$ a_2=1 $$ and $$ \mu_1=\mu_2=\frac{11}{10} $$ , which satisfy (7) and (8). A direct computation yields

and

Then, we obtain

which means that the condition (9) holds with $$ \lambda_1=-\frac{407}{112} $$ , $$ \lambda_2=\frac{169}{112} $$ , $$ \lambda_3=\frac{671}{252} $$ , $$ \lambda_4=\frac{67}{56} $$ , $$ W(x)=|x|^8 $$ , $$ W(y)=|y|^8 $$ , and $$ q=6 $$ . Let $$ \mathcal{J}_{a1}=\mathcal{J}_{a2}=1s $$ (i.e., the active period of each subsystem is 1 s) and $$ N_{01}=N_{02}=0.1 $$ . From (11) and (12), we can compute that the constant $$ \varepsilon $$ should satisfy $$ 0.0953<\varepsilon<0.7883 $$ . Then, it follows from (10) that $$ \varepsilon=0.4414 $$ . According to Theorem 1, the highly nonlinear SSS (30) has a unique global solution on $$ [-\frac{3}{4}, \infty) $$ and is bounded. In addition, the system (30) is not only $$ 6 $$ th moment exponentially stable but also almost surely exponentially stable. Figure 2 shows that the system (30) is exponentially stable in $$ 6 $$ th moment. Figure 3 shows that the system (30) is exponentially stable in the sample path. Figure 4 shows the switching signal $$ \sigma(t) $$ .

Figure 2. The exponential stability in $$ L^6 $$ of system (30).

Figure 3. Exponential stability in the sample path of the system (30).

Figure 4. Switching signal $$ \sigma(t) $$ .

5. CONCLUSIONS

In this paper, the existence of a unique global solution for a highly nonlinear SSS with a deterministic switching signal is examined by using the ADT method coupled with the MLF approach. The stability criteria of $$ q $$ th moment exponential stability and almost surely exponential stability of the highly nonlinear SSS are stated. Finally, a numerical example is presented to illustrate the effectiveness of the obtained results. Inspired by recent studies ^{[7, 20, 33-36]}, two further research directions have emerged: (1) Solving the problem of stability for highly nonlinear SSSs with impulsive effects under asynchronous switching, and (2) designing a control input function to stabilize a highly nonlinear SSS with a time-varying delay.

DECLARATIONS