Robust Stability Analysis of Switched Neural Networks with Application in Psychological Counseling Evaluation System

Abstract

In this work, the effectiveness and stability of psychological counseling are evaluated using the switched complex-valued neural networks (SCVNN) model, which includes parameter disturbances, impulsive perturbations, variable and continuously distributed delays in the system state, and impulsive delay. How to analyze and judge the stability of the network simply and effectively is the primary prerequisite for its successful application. Therefore, we explore the dynamic behavior of SCVNN with both variable and distributed delays along with impulsive effect. Initially, the proposed conditions for the existence and uniqueness of equilibrium in SCVNN are presented. Subsequently, employing the inequality technique and impulsive average dwell time approach, sufficient conditions for the robust exponential stability of SCVNN under both arbitrary and restricted switching are obtained. Lastly, the psychological counseling evaluation system (PCES) is established, and a simulation example is used to verify the correctness and effectiveness of the presented findings.

1. Introduction

Neural networks have recently been utilized for the assessment and analysis of mental health and psychological counseling [1,2,3]. In [3], a method of emotional analysis based on the CNN–BiLSTM hybrid neural network has been developed. As shown in Figure 1, suppose that the neural networks have n neurons, denoted by $x_1,x_2,\cdots ,x_n$, which, respectively, represent n evaluation indexes in the PCES. The link weight between any two neurons $x_i$ and $x_j$ in the networks is denoted as $a_{ij}$. To assess the effectiveness of psychological counseling, we should first evaluate the evaluation indicators and evaluation objectives according to the evaluation indicator standards of psychological counseling and form sample data. Then, the connection weights of neurons will be determined according to the relationship between evaluation indexes. Psychological counseling is a process, and the effectiveness of counseling also needs a process of accumulation. The previous counseling is the stage of collecting information and establishing relationships. Psychological counseling usually takes effect after several consecutive efforts. And every counseling is affected by some subjective and objective factors, so the correlation weights of evaluation indicators are generally different. As is widely recognized, the connections between neurons in neural networks are often unstable, leading to link failures and the formation of new connections. Consequently, sudden changes in neural network architectures and parameters frequently occur, causing switching between various topologies. Therefore, evaluation of the whole psychological counseling must use the switched network system, as shown in the Figure 2. Switched neural networks are hybrid systems that fundamentally consist of multiple subnetworks and a switching signal, which designates a specific subnetwork to be activated at any given time. The properties of a switched system are much more complex than those of each individual subsystem. For instance, even if all subsystems are stable, an unconstrained switching signal may destabilize the entire system. Conversely, even if all subsystems are unstable, an appropriate switching signal can stabilize the system. Therefore, studying of the stability, controller design, and switching signal design for switched systems are challenging tasks that present significant difficulties. It is shown that the application of neural networks depends greatly on their dynamic behavior [4]. For example, in the creation of associative memory for neural networks, refs. [5,6] require that the state of each neuron converge to the network’s unique equilibrium point. Consequently, the exploration of neural network stability carries significant theoretical and practical implications. In [7], the exponential stability of switched neural networks is evaluated by combining the dwell time technique with discretized Lyapunov–Krasovskii functions. The Lyapunov function approach, the comparison principle, and an impulsive delay differential inequality are used to investigate the synchronization of coupled switching neural networks [8].

Complex-valued neural networks (CVNN) are extremely important to research since they are widely used in many domains, including signal processing, remote sensing, voice synthesis, filtering, and others, and they cannot be fully evaluated using simply their real-valued counterparts [9,10,11,12,13]. Due to their more complex characteristics and methodologies for analyzing dynamic behaviors, such as synchronization, stability, and periodicity of system states, CVNN are not merely straightforward extensions of real-valued systems. Consequently, a multitude of significant studies has emerged over recent years, delving into the analysis of dynamic behavior in CVNN. The study in [14] investigated the finite-/fixed-time synchronization of CVNN with diffusion and discontinuities. The study in [15] explored the finite-time stability of a specific category of CVNN with fractional order. Using integral inequalities, the robust stability of stochastic CVNN was investigated in [16]. Nevertheless, the models discussed in the aforementioned references did not incorporate a switching signal.

In actuality, time delays in the electrical implementation of neural networks are unavoidable. These delays typically lead to a degradation in model performance, and as the delays increase, the stability of the models may even be compromised [17]. Neural networks typically feature spatial aspects because of the numerous parallel paths with varying axon sizes and lengths. In light of this, it becomes necessary to apply continually distributed delays over a predetermined period of time to reflect the fact that actions from remote past states are less significant than those from the present [18,19]. In real-world neural networks, the impulsive impact is unavoidable. For example, system states are impulsively affected by instantaneous perturbations, which might be brought on by a sudden noise or a change in frequency. Exponential convergence of time-varying delayed T-S fuzzy CVNN with impulsive effects was explored in [20]. Exponential stability of time-delayed impulsive CVNN was examined in [21]. Furthermore, certain parameters cannot be precisely specified due to a lack of knowledge and information. Additionally, various factors, including modeling errors and external perturbations, are inescapable. As a result, the models inherently possess uncertainty, leading to a significant influence on the dynamic behaviors of the systems. Assuming the parameters are contained inside a specific interval is an acceptable way to assess the robustness of the models [22].

Inspired by the previous explanation, an investigation into the robust stability of a class of SCVNN featuring variable and distributed delays together with impulsive impact on the field $\mathbb{C}$ is planned. The contributions of this paper are as follows: (1) In this work, robust stability of the SCVNN under arbitrary switching is studied. Addressing robust stability under arbitrary switching poses substantial theoretical challenges, necessitating innovative approaches in Lyapunov function construction, system modeling, and uncertainty management. Overcoming these challenges not only refines the existing theoretical framework but also advances stability theories for switched systems, offering new perspectives and methods for future research. (2) In the stability analysis of the system, more influential factors are taken into account (including parameter disturbances and impulsive perturbations, variable and continuously distributed delays of system state, and impulsive delay), which makes the model more close to the actual operation conditions and broadens its applicability. (3) The results obtained using the LMI (linear matrix inequality) approach often require manual determination of certain parameters. In contrast, the sufficient conditions proposed in this paper are algebraic expressions, making them more intuitive and advantageous for practical applications.

Notations: Let the real number set be represented by $\mathbb{R}$ and the complex number set by $\mathbb{C}$. Let $z=x+y\mathrm{i}\in \mathbb{C}$, where ‘i’ represents imaginary unit and $x,y\in \mathbb{R}$. Indicate by $\left|z\right|\triangleq \sqrt{x^2+y^2}$ the module of z and by $\overline{z}\triangleq x-y\mathrm{i}$ its conjugate complex number, respectively, assuming that the column vector of ${\mathbb{C}}^n$ is $\mathit{z}={\left(z_1,z_2,\dots ,z_n\right)}^{\mathrm{T}}$, where ‘T’ stands for transposition. The vector norm denoted by $\parallel \mathit{z}\parallel =\sqrt{{\sum }_{j=1}^n|z_j{|}^2}$ and $\left|\mathit{z}\right|$ represents ${\left(\left|z_1\right|,\left|z_2\right|,\dots ,\left|z_n\right|\right)}^{\mathrm{T}}$. $\left|\mathit{C}\right|$ represents $\left(\right|c_{ij}{\left|\right)}_{n\times n}\in {\mathbb{R}}^{n\times n}$ for the complex number matrix $\mathit{C}={\left(c_{ij}\right)}_{n\times n}\in {\mathbb{C}}^{n\times n}$. Re(·) and Im(·) are the real component and imaginary component of a complex number, respectively.

2. Preliminaries

The SCVNN model can be formulated through the utilization of delayed differential equations in the following:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\tilde{z}}_i\left(t\right)}{\mathrm{d}t}=& {\tilde{h}}_i^{\sigma \left(t\right)}\left({\tilde{z}}_i\left(t\right)\right)\left\{-{\tilde{d}}_i^{\sigma \left(t\right)}\left({\tilde{z}}_i\left(t\right)\right)+\sum _{j=1}^n[a_{ij}^{\sigma \left(t\right)}{g}_j^{\sigma \left(t\right)}\left({\tilde{z}}_j\left(t\right)\right)\right.\hfill \\ & +b_{ij}^{\sigma \left(t\right)}{g}_j^{\sigma \left(t\right)}\left({\tilde{z}}_j\left(t-{\tau }_{ij}^{\sigma \left(t\right)}\left(t\right)\right)\right)\hfill \\ & +c_{ij}^{\sigma \left(t\right)}{\int }_{-\infty }^t{\chi }_{ij}^{\sigma \left(t\right)}\left(t-s\right)g_j^{\sigma \left(t\right)}\left({\tilde{z}}_j\left(s\right)\right)\mathrm{d}s+I_i^{\sigma \left(t\right)}\left]\right\},t\ne t_k,\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill {\tilde{z}}_i\left(t_k\right)=& {\tilde{\varphi }}_{1i}^k\left({\tilde{z}}_1\left(t_k^{-}\right),\cdots ,{\tilde{z}}_n\left(t_k^{-}\right)\right)+{\tilde{\varphi }}_{2i}^k\left({\tilde{z}}_1\left({\left(t_k-{\tau }_{i1}\left(t_k\right)\right)}^{-}\right),\cdots ,{\tilde{z}}_n\left({\left(t_k-{\tau }_{in}\left(t_k\right)\right)}^{-}\right)\right)+J_{ik},\hfill \end{array}$$

(1b)

where $i=1,2,\cdots ,n$, $k=1,2,\cdots$. ${\tilde{z}}_i\left(t\right)\in \mathbb{C}$ represents the state of neuron i. ${\tilde{h}}_i^{\sigma \left(t\right)}$, ${\tilde{d}}_i^{\sigma \left(t\right)}$, and $g_i^{\sigma \left(t\right)}$ are amplification functions, self-feedback functions, and activation functions that depend on $\sigma \left(t\right)$, respectively. $\sigma \left(t\right)$ is a right-continuous function that maps from $\left[0,+\infty \right)$ to the set $\Xi \triangleq \left\{1,2,\cdots ,m\right\}$, exhibiting a piecewise constant nature. $\sigma \left(t\right)$ is called a switching law or switching signal, and $\sigma \left(t\right)=q\in \Xi$ indicates the activation of the qth subnetwork. ${\mathit{A}}_q={\left(a_{ij}^q\right)}_{n\times n}\in {\mathbb{C}}^{n\times n}$, ${\mathit{B}}_q={\left(b_{ij}^q\right)}_{n\times n}\in {\mathbb{C}}^{n\times n}$, and ${\mathit{C}}_q={\left(c_{ij}^q\right)}_{n\times n}\in {\mathbb{C}}^{n\times n}$ are complex-valued matrices representing the connection weights of the qth subnetwork. ${\chi }_{ij}^q:\left[0,+\infty \right)\to \left[0,+\infty \right)$ is the kernel function of the qth subnetwork satisfy ${\int }_0^{\infty }{e}^{\vartheta s}{\chi }_{ij}^q\left(s\right)\mathrm{d}s={{\rm Y}}_{ij}^q\left(\vartheta \right)$, where ${{\rm Y}}_{ij}^q\left(0\right)=1$ and ∀$\delta >0$, ${{\rm Y}}_{ij}^q\left(\vartheta \right)$ is continuous on $\left[0,\delta \right)$. ${\mathit{I}}_q={\left(I_1^q,I_2^q,\cdots ,I_n^q\right)}^{\mathrm{T}}\in {\mathbb{C}}^n$ is a constant external input vector of the qth subnetwork. ${\tau }_{ij}\left(t_k\right)$ and ${\tau }_{ij}^q\left(t\right)$ are bounded delay function with $0\le {\tau }_{ij}\left(t_k\right)\le \tau$ and $0\le {\tau }_{ij}^q\left(t\right)\le {\tau }^q$, respectively. Equation (1b) describes the sudden alteration in the state occurring at $t_k$, where ${\tilde{\varphi }}_{1i}\left({\tilde{z}}_1\left(t_k^{-}\right),\cdots ,{\tilde{z}}_n\left(t_k^{-}\right)\right)$ is the impulsive perturbation of neuron i at $t_k$ with ${\tilde{z}}_i\left(t_k^{-}\right)={lim}_{t\to t_k^{-}}{\tilde{z}}_i\left(t\right)$; ${\tilde{\varphi }}_{2i}({\tilde{z}}_1\left({\left(t_k-{\tau }_{i1}\left(t_k\right)\right)}^{-}\right),\cdots ,$${\tilde{z}}_n\left({\left(t_k-{\tau }_{in}\left(t_k\right)\right)}^{-}\right))$ is the impulsive perturbation of neuron i at $t_k$, which is triggered by transmission delays; and $J_{ik}$ is impulsive inputs at $t_k$. ${\tilde{z}}_i\left(s+t_0\right)={\tilde{\phi }}_i\left(s\right)$ is taken to be the initial condition of Equations (1a) and (1b), here ${\tilde{\phi }}_i\in C\left(\left(-\infty ,t_0\right],\mathbb{C}\right),i=1,2,\cdots ,n$.

In the process of psychological counseling, the effect of counseling is subject to uncertain external disturbance. So $\sigma \left(t\right)$ is a function of time that is unknown in advance. Under the assumption that the $i_p$th subnetwork is turned on when $t\in \left(t_p,t_{p+1}\right]$, we can derive the switching sequence $\left\{\left(t_0,i_0\right),\cdots ,\right.$$\left.\left(t_p,i_p\right),\cdots |i_p\in \Sigma ,p=0,1,\cdots \right\}$. We further assume that there exists only a finite number of switches within any finite interval, and that the following conditions are met.

Assumption 1 

([23]). For any $u\left(t\right),v\left(t\right)\in \mathbb{C}$, positive numbers ${\mathcal{D}}_i^q$$(q\in \Xi ;i=1,2,\cdots ,n)$ exist such that

$$\frac{{\tilde{d}}_i^q\left(u\left(t\right)\right)-{\tilde{d}}_i^q\left(v\left(t\right)\right)}{u\left(t\right)-v\left(t\right)}\ge {\mathcal{D}}_i^q$$

holds. Let ${\mathcal{D}}_q=\mathrm{diag}\left({\mathcal{D}}_1^q,{\mathcal{D}}_2^q,\cdots ,{\mathcal{D}}_n^q\right).$

Assumption 2. 

For any $u\left(t\right),v\left(t\right)\in \mathbb{C}$, positive numbers ${\mathcal{L}}_i^q$$(q\in \Xi ;i=1,2,\cdots ,n)$ exist such that

$$|g_i^q\left(u\left(t\right)\right)-g_i^q\left(v\left(t\right)\right)|\le {\mathcal{L}}_i^q\left|u\left(t\right)-v\left(t\right)\right|$$

holds. Let ${\mathcal{L}}_q=\mathrm{diag}\left({\mathcal{L}}_1^q,{\mathcal{L}}_2^q,\cdots ,{\mathcal{L}}_n^q\right)$.

Assumption 3. 

For amplification function ${\tilde{h}}_i^q\left({\tilde{z}}_i\left(t\right)\right)$, real numbers ${\mathcal{H}}_i^q>0$ $(q\in \Xi ;i=1,2,\cdots ,n)$ exist such that ${\tilde{h}}_i^q\left({\tilde{z}}_i\left(t\right)\right)\ge {\mathcal{H}}_i^q>0$ holds.

Remark 1. 

Assumptions 1–3 are general conditions (see, e.g., [23,24,25,26,27,28,29,30,31]). In the dynamic behavior analysis of CVNN, there are many studies, such as [24,25,26,27,28,29,30], which must explicitly decompose the activation function, amplification function, and self feedback function into real part and imaginary part. And they also assume that these functions are differentiable and bounded. It is crucial to acknowledge, however, that these divisions are not always expressible in an analytical sense [23,31]. In this work, the activation function simply needs to fulfill the Lipschitz condition, and the amplification function only needs to have a lower bound.

Assumption 4. 

There are non-negative square matrices ${\tilde{\mathit{\Phi }}}_{1k}=\left({\tilde{\phi }}_{ij}^{1k}\right)$ and ${\tilde{\mathit{\Phi }}}_{2k}=\left({\tilde{\phi }}_{ij}^{2k}\right)$ of order n over $\mathbb{R}$, such that for any ${\mathit{z}}_1={(z_{11},\cdots ,z_{1n})}^{\mathrm{T}}$ and ${\mathit{z}}_2={(z_{21},\cdots ,z_{2n})}^{\mathrm{T}}\in {\mathbb{C}}^n$,

$$\left|{\tilde{\varphi }}_{1i}^k\left(z_{11},\cdots ,z_{1n}\right)-{\tilde{\varphi }}_{1i}^k\left(z_{21},\cdots ,z_{2n}\right)\right|\le \sum _{j=1}^n{\tilde{\phi }}_{ij}^{1k}\left|z_{1j}-z_{2j}\right|,$$

$$\left|{\tilde{\varphi }}_{2i}^k\left(z_{11},\cdots ,z_{1n}\right)-{\tilde{\varphi }}_{2i}^k\left(z_{21},\cdots ,z_{2n}\right)\right|\le \sum _{j=1}^n{\tilde{\phi }}_{ij}^{2k}\left|z_{1j}-z_{2j}\right|$$

hold, where $i=1,2,\cdots ,n$ and $k=1,2,\cdots$.

Remark 2. 

The impulsive phenomena is assumed to exclusively occur in switching instants for the sake of simplicity in the analysis that follows. Indeed, the impulse disturbance can be regarded as a form of system switching. However, during these specific switching moments, the subsystems remain unchanged.

Considering the parameter uncertainty in (1a) and (1b), the interval weight matrices for every subnetwork q, $q\in \Xi$ are constructed as follows:

$$\begin{array}{c}\hfill {\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}=\left\{|{\mathit{A}}_q|={\left(\right|a_{ij}^q\left|\right)}_{n\times n}:|{\underset{\sim }{\mathit{A}}}_q|\le |{\mathit{A}}_q|\le |{\tilde{\mathit{A}}}_q|,\mathrm{i}.\mathrm{e}.,|{\underset{\sim }{a}}_{ij}^q|\le |a_{ij}^q|\le |{\tilde{a}}_{ij}^q|,i,j=1,2,\cdots ,n\right\},\\ \hfill {\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}=\left\{|{\mathit{B}}_q|={\left(\right|b_{ij}^q\left|\right)}_{n\times n}:|{\underset{\sim }{\mathit{B}}}_q|\le |{\mathit{B}}_q|\le |{\tilde{\mathit{B}}}_q|,\mathrm{i}.\mathrm{e}.,|{\underset{\sim }{b}}_{ij}^q|\le |b_{ij}^q|\le |{\tilde{b}}_{ij}^q|,i,j=1,2,\cdots ,n\right\},\\ \hfill {\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}=\left\{|{\mathit{C}}_q|={\left(\right|c_{ij}^q\left|\right)}_{n\times n}:|{\underset{\sim }{\mathit{C}}}_q|\le |{\mathit{C}}_q|\le |{\tilde{\mathit{C}}}_q|,\mathrm{i}.\mathrm{e}.,|{\underset{\sim }{c}}_{ij}^q|\le |c_{ij}^q|\le |{\tilde{c}}_{ij}^q|,i,j=1,2,\cdots ,n\right\},\end{array}$$

(2)

where ${\underset{\sim }{a}}_{ij}^q$, ${\underset{\sim }{b}}_{ij}^q$, ${\underset{\sim }{c}}_{ij}^q$, ${\tilde{a}}_{ij}^q$, ${\tilde{b}}_{ij}^q$, and ${\tilde{c}}_{ij}^q$ are all known complex value constants.

Definition 1. 

Let ${\mathit{z}}^{\ast }$ be the equilibrium point of (1a) and (1b). For ${\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}$, ${\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}$, and ${\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}$ ($q\in \Xi$), if $\lambda >0$ and $M>1$ are constants such that

$$∥\mathit{z}\left(t\right)-{\mathit{z}}^{\ast }∥\le M{sup}_{s\in \left(-\infty ,t_0\right]}∥\mathit{\varphi }\left(s\right)-{\mathit{z}}^{\ast }∥exp(-\lambda (t-t_0)),t\ge t_0,$$

then ${\mathit{z}}^{\ast }$ is considered to be robustly exponentially stable.

Lemma 1 

([32]). Consider a square matrix $\mathit{B}$ of order n over the field of real numbers, which has non-positive off-diagonal entries. The assertions that follow are synonymous:

1. 

$\mathit{B}$ is an M-matrix;

2. 

A positive real vector $\mathit{\xi }$ exists such that $\mathit{B}\mathit{\xi }>\mathbf{0}\left({\mathit{\xi }}^{\mathrm{T}}\mathit{B}>\mathbf{0}\right)$.

Lemma 2 

([23,33]). If $\mathcal{F}\left(\mathit{z}\right)$ is a continuous function over the complex n-space ${\mathbb{C}}^n$ that fulfills both of the following two requirements:

1. 

$\mathcal{F}\left(\mathit{z}\right)$ is univalent injective on ${\mathbb{C}}^n$;

2. 

$∥\mathcal{F}\left(\mathit{z}\right)∥\to \infty$ as $∥\mathit{z}∥\to \infty$,

then $\mathcal{F}\left(\mathit{z}\right)$ is a homeomorphism mapping form ${\mathbb{C}}^n$ onto itself.

3. Main Results

By leveraging the distinctive properties of M-matrix and homeomorphism, it will first be demonstrated that each subnetwork q ($q\in \Xi$) has an equilibrium point ${\mathit{z}}_q^{\ast }$, and that each point is unique, implying that (1a) and (1b) has an equilibrium point.

Theorem 1. 

It is supposed that the Assumptions mentioned above are satisfied. If for any $q\in \Xi$, ${\mathit{D}}_q={\left(d_{ij}^q\right)}_{n\times n}$ are nonsingular M-matrices, here, $d_{ii}^q={\mathcal{D}}_i^q$ and $d_{ij}^q=-{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^q|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right),i\ne j$. Then, for ${\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}$, ${\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}$, ${\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}$ and every given switching signal $\sigma \left(t\right)$, (1a) and (1b) possesses a unique equilibrium point.

Proof of Theorem 1. 

Define a map

$${\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)={\left({\mathcal{F}}_1^q\left(\tilde{\mathit{z}}\left(t\right)\right),{\mathcal{F}}_2^q\left(\tilde{\mathit{z}}\left(t\right)\right),\cdots ,{\mathcal{F}}_n^q\left(\tilde{\mathit{z}}\left(t\right)\right)\right)}^{\mathrm{T}},$$

associated with (1a) and (1b) as the forms:

$${\mathcal{F}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)=-{\tilde{d}}_i^q\left({\tilde{z}}_i\left(t\right)\right)+\sum _{j=1}^n\left(a_{ij}^q+b_{ij}^q+c_{ij}^q\right)g_j^q\left({\tilde{z}}_j\left(t\right)\right)+I_i^q.$$

(3)

As is widely known, if for all $q\in \Xi$, ${\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)$ are homeomorphism maps on ${\mathbb{C}}^n$, then each subnetwork of (1a) and (1b) possesses a distinct equilibrium point [34].

Initially, we demonstrate that ${\mathcal{F}}^q\left(\tilde{\mathit{z}}\left(t\right)\right)$ are univalent injective mappings on ${\mathbb{C}}^n$. Effectively, if there are vectors ${\tilde{\mathit{z}}}^{\prime }$ and ${\tilde{\mathit{z}}}^{″}$ over the complex n-space ${\mathbb{C}}^n$ and ${\tilde{\mathit{z}}}^{\prime }\ne {\tilde{\mathit{z}}}^{″}$ such that

$$\begin{array}{cc}\hfill -& {\tilde{d}}_i^q\left({\tilde{z}}_i^{\prime }\left(t\right)\right)+\sum _{j=1}^n\left(a_{ij}^q+b_{ij}^q+c_{ij}^q\right)g_j^q\left({\tilde{z}}_j^{\prime }\left(t\right)\right)+I_i^q\hfill \\ \hfill =-& {\tilde{d}}_i^q\left({\tilde{z}}_i^{″}\left(t\right)\right)+\sum _{j=1}^n\left(a_{ij}^q+b_{ij}^q+c_{ij}^q\right)g_j^q\left({\tilde{z}}_j^{″}\left(t\right)\right)+I_i^q,\hfill \end{array}$$

then considering Assumptions 1 and 2, as well as (1a) and (1b), we arrive at

$${\mathcal{D}}_i^q\left|{\tilde{z}}_i^{\prime }-{\tilde{z}}_i^{″}\right|-\sum _{j=1}^n{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^p|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right)\left|{\tilde{z}}_j^{\prime }-{\tilde{z}}_j^{″}\right|\le 0,i=1,2,\cdots ,n.$$

(4)

It is clear that inequalities (4) can be written in vector form:

$${\mathit{D}}_q|{\tilde{\mathit{z}}}^{\prime }-{\tilde{\mathit{z}}}^{″}|\le \mathbf{0}.$$

(5)

From ${\mathit{D}}_q$, a nonsingular M-matrix, it follows that det ${\mathit{D}}_q>0$ and ${\mathit{D}}_q^{-1}>0$ exists. One may determine that $|{\tilde{\mathit{z}}}^{\prime }-{\tilde{\mathit{z}}}^{″}|=0$. This is in conflict with ${\tilde{\mathit{z}}}^{\prime }\ne {\tilde{\mathit{z}}}^{″}$. Therefore, for all $q\in \Xi ,{\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)$ are univalent injective on ${\mathbb{C}}^n$.

We shall demonstrate in the following that $∥{\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)∥\to \infty$ as $∥\tilde{\mathit{z}}\left(t\right)∥\to \infty$. Let ${\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)={\mathcal{F}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)-{\mathcal{F}}_i^q\left(\mathbf{0}\right)$, i.e.,

$${\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)=-\left[{\tilde{d}}_i^q\left({\tilde{z}}_i\left(t\right)\right)-{\tilde{d}}_i^q\left(0\right)\right]+\sum _{j=1}^n\left(a_{ij}^q+b_{ij}^q+c_{ij}^q\right)\left[g_j^q\left({\tilde{z}}_j\left(t\right)\right)-g_j^q\left(0\right)\right],$$

(6)

where $q\in \Xi ,i=1,2,\cdots ,n$. Taking Assumptions 1 and 2 and (1a) and (1b) into account, and multiplying both sides of (6) by ${\overline{\tilde{z}}}_i$, we arrive at

$$\begin{array}{cc}\hfill \mathrm{Re}\left({\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right){\overline{\tilde{z}}}_i\left(t\right)\right)=& -\mathrm{Re}\left\{\left[{\tilde{d}}_i^q\left({\tilde{z}}_i\left(t\right)\right)-{\tilde{d}}_i^q\left(0\right)\right]{\overline{\tilde{z}}}_i\left(t\right)\right\}\hfill \\ \hfill & +\mathrm{Re}\left\{{\overline{\tilde{z}}}_i\left(t\right)\sum _{j=1}^n\left(a_{ij}^q+b_{ij}^q+c_{ij}^q\right)\left[g_j^q\left({\tilde{z}}_j\left(t\right)\right)-g_j^q\left(0\right)\right]\right\}\hfill \\ \hfill \le & -{\mathcal{D}}_i^q|{\tilde{z}}_i\left(t\right){|}^2+\left|{\overline{\tilde{z}}}_i\left(t\right)\right|\sum _{j=1}^n{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^q|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right)\left|{\tilde{z}}_j\left(t\right)\right|.\hfill \end{array}$$

(7)

As we are aware, there exist positive matrices ${\mathit{P}}_q=\mathrm{diag}\{{\xi }_1^q,{\xi }_2^q,\cdots ,{\xi }_n^q\}$, $q\in \Xi$ such that ${\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q$ are positive definite matrices based on the criteria of Theorem 1. Multiplying by ${\xi }_i^q$ on both sides of (7) and taking the sum, we obtain

$$\begin{array}{cc}\hfill & \sum _{i=1}^n{\xi }_i^q\mathrm{Re}\left({\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right){\overline{\tilde{z}}}_i\left(t\right)\right)\hfill \\ \hfill \le & \sum _{i=1}^n{\xi }_i^q\left[-{\mathcal{D}}_i^q|{\tilde{z}}_i\left(t\right){|}^2+\left|{\overline{\tilde{z}}}_i\left(t\right)\right|\sum _{j=1}^n{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^q|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right)\left|{\tilde{z}}_j\left(t\right)\right|\right]\hfill \\ \hfill =& -\left(\left|z_1\left(t\right)\right|,\left|z_2\left(t\right)\right|,\cdots ,\left|z_n\left(t\right)\right|\right)P_q{D}_q{\left(\left|z_1\left(t\right)\right|,\left|z_2\left(t\right)\right|,\cdots ,\left|z_n\left(t\right)\right|\right)}^{\mathrm{T}}\hfill \\ \hfill =& -\frac{1}{2}{\left|\mathit{z}\left(t\right)\right|}^{\mathrm{T}}\left({\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q\right)\left|\mathit{z}\left(t\right)\right|\hfill \\ \hfill \le & -\frac{1}{2}{\lambda }_{min}\left({\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q\right){∥\mathit{z}\left(t\right)∥}^2,\hfill \end{array}$$

(8)

i.e.,

$$\begin{array}{cc}\hfill {\lambda }_{min}\left({\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q\right){∥\mathit{z}\left(t\right)∥}^2\le & -2\sum _{i=1}^n{\xi }_i^q\mathrm{Re}\left({\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right){\overline{\tilde{z}}}_i\left(t\right)\right)\hfill \\ \hfill \le & {\xi }_{\max }^q\sum _{i=1}^n\left|{\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)\right|\left|{\overline{\tilde{z}}}_i\left(t\right)\right|\hfill \\ \hfill \le & {\xi }_{\max }^q\sum _{i=1}^n{\left|{\tilde{\mathcal{F}}}_i^q\left(\tilde{\mathit{z}}\left(t\right)\right)\right|}^2+{\xi }_{\max }^q\sum _{i=1}^n{\left|{\overline{\tilde{z}}}_i\left(t\right)\right|}^2,\hfill \end{array}$$

(9)

where ${\xi }_{max}^q={max}_{1\le i\le n}\left\{{\xi }_i\right\}$ and ${\lambda }_{min}\left({\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q\right)$ represents the minimum eigenvalue of ${\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q$. It follows from (9) that

$$\parallel \tilde{\mathit{z}}\left(t\right){\parallel }^2\le {\left({\lambda }_{min}\left({\mathit{P}}_q{\mathit{D}}_q+{\mathit{D}}_q^{\mathrm{T}}{\mathit{P}}_q\right)-{\xi }_{max}^q\right)}^{-1}{\xi }_{max}^q\parallel {\tilde{\mathcal{F}}}_q\left(\tilde{\mathit{z}}\left(t\right)\right){\parallel }^2.$$

(10)

Obviously, it follows from (10) that $∥{\tilde{\mathcal{F}}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)∥\to \infty$ as $∥\tilde{\mathit{z}}\left(t\right)∥\to \infty$, which implies that $∥{\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)∥\to \infty$ as $∥\tilde{\mathit{z}}\left(t\right)∥\to \infty$. In light of Lemma 2, for all $q\in \Xi$, ${\mathcal{F}}_q\left(\tilde{\mathit{z}}\left(t\right)\right)$ are homeomorphism maps on ${\mathbb{C}}^n$. Therefore, every subnetwork possesses a unique equilibrium point. Further, we can determine that (1a) and (1b) has a unique equilibrium point for each given switching signal. □

Let ${\mathit{z}}^{\ast }={(z_1^{\ast },z_2^{\ast },\cdots ,z_n^{\ast })}^{\mathrm{T}}$ and $\tilde{\mathit{z}}\left(t\right)={({\tilde{z}}_1\left(t\right),{\tilde{z}}_2\left(t\right),\cdots ,{\tilde{z}}_n\left(t\right))}^{\mathrm{T}}$ be an equilibrium point and any solution of (1a) and (1b), respectively. Denote

$$\begin{array}{c}{z}_i\left(t\right)={\tilde{z}}_i\left(t\right)-z_i^{\ast },h_i^q\left(z_i\left(t\right)\right)={\tilde{h}}_i^q\left(z_i\left(t\right)+z_i^{\ast }\right),d_i^q\left(z_i\left(t\right)\right)={\tilde{d}}_i^q\left(z_i\left(t\right)+z_i^{\ast }\right)-{\tilde{d}}_i^q\left(z_i^{\ast }\right),\hfill \\ f_j^q\left(z_j\left(t\right)\right)=g_j^q(z_j\left(t\right)+z_j^{\ast })-g_j^q\left(z_j^{\ast }\right),\hfill \\ {\varphi }_{1i}^k\left(z_1\left(t_k\right),\cdots ,z_n\left(t_k\right)\right)={\tilde{\varphi }}_{1i}^k\left(z_1\left(t_k\right)+z_1^{\ast },\cdots ,z_n\left(t_k\right)+z_n^{\ast }\right)-{\tilde{\varphi }}_{1i}^k\left(z_1^{\ast },\cdots ,z_n^{\ast }\right),\hfill \\ {\varphi }_{2i}^k\left(z_1\left(t_k-{\tau }_{i1}\left(t_k\right)\right),\cdots ,z_n\left(t_k-{\tau }_{in}\left(t_k\right)\right)\right)=\hfill \\ {\tilde{\varphi }}_{2i}^k\left(z_1\left(t_k-{\tau }_{i1}\left(t_k\right)\right)+z_1^{\ast },\cdots ,z_n\left(t_k-{\tau }_{in}\left(t_k\right)\right)+z_n^{\ast }\right)-{\tilde{\varphi }}_{2i}^k\left(z_1^{\ast },\cdots ,z_n^{\ast }\right),\hfill \end{array}$$

then, (1a) and (1b) is readily converted to the forms as follows:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{z}_i\left(t\right)}{\mathrm{d}t}=& h_i^{\sigma \left(t\right)}\left(z_i\left(t\right)\right)\left\{-d_i^{\sigma \left(t\right)}\left(z_i\left(t\right)\right)+\sum _{j=1}^n[a_{ij}^{\sigma \left(t\right)}{f}_j^{\sigma \left(t\right)}\left(z_j\left(t\right)\right)\right.\hfill \\ & +b_{ij}^{\sigma \left(t\right)}{f}_j^{\sigma \left(t\right)}\left(z_j\left(t-{\tau }_{ij}^{\sigma \left(t\right)}\left(t\right)\right)\right)\hfill \\ & +c_{ij}^{\sigma \left(t\right)}{\int }_{-\infty }^t{\chi }_{ij}^{\sigma \left(t\right)}\left(t-s\right)f_j^{\sigma \left(t\right)}\left(z_j\left(s\right)\right)\mathrm{d}s\left]\right\},t\ne t_k,\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill z_i\left(t_k\right)=& {\varphi }_{1i}^k\left(z_1\left(t_k^{-}\right),\cdots ,z_n\left(t_k^{-}\right)\right)+{\varphi }_{2i}^k\left(z_1\left({\left(t_k-{\tau }_{i1}\left(t_k\right)\right)}^{-}\right),\cdots ,z_n\left({\left(t_k-{\tau }_{in}\left(t_k\right)\right)}^{-}\right)\right),\hfill \end{array}$$

(11b)

where $i=1,2,\cdots ,n$, $k=1,2,\cdots$. Let ${\phi }_i\left(s\right)={\tilde{\phi }}_i\left(s\right)-z_i^{\ast }$ be the initial condition of (11a) and (11b).

Define the indicator function $\mathit{\varsigma }\left(t\right)={\left({\varsigma }_1\left(t\right),{\varsigma }_2\left(t\right),\cdots ,{\varsigma }_m\left(t\right)\right)}^{\mathrm{T}}$, where

$${\varsigma }_q\left(t\right)=\left\{\begin{array}{c}1,\mathrm{when}\mathrm{the}q-\mathrm{th}\mathrm{subnetworks}\mathrm{is}\mathrm{activated};\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Therefore, (11a) and (11b) without impulsive effect is converted to the form:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{z}_i\left(t\right)}{\mathrm{d}t}=& \sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)\{-d_i^q\left(z_i\left(t\right)\right)+\sum _{j=1}^n[a_{ij}^q{f}_j^q\left(z_j\left(t\right)\right)+b_{ij}^q{f}_j^q\left(z_j\left(t-{\tau }_{ij}^q\left(t\right)\right)\right)\hfill \\ & +c_{ij}^q{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)f_j^q\left(z_j\left(s\right)\right)\mathrm{d}s\left]\right\},i=1,2,\cdots ,n.\hfill \end{array}$$

(12)

Lemma 3. 

It is assumed that Assumptions 1–3 are true. If $\mathit{\xi }={({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)}^{\mathrm{T}}$ is a positive vector over the real n-space ${\mathbb{R}}^n$, and $\lambda$ is a positive real number, such that for all $q\in \Xi$, inequalities

$$\left(\frac{\lambda }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j\left(|a_{ij}^q|+exp\left(\frac{\lambda }{2}{\tau }^q\right)|b_{ij}^q|+\left|c_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{\lambda }{2}\right)\right)<0$$

(13)

hold; then, for ${\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}$, ${\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}$, and ${\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}$, the equilibrium point of (12) is robustly exponentially stable under arbitrary switching.

Proof of Lemma 3. 

Choose a candidate vector Lyapunov function as depicted below:

$$V_i(t,z_i\left(t\right))=exp\left(\lambda \left(t-t_0\right)\right){\left|z_i\left(t\right)\right|}^2.$$

(14)

After taking Assumptions 1–3 into account and computing the upper right derivative of $V_i(t,z_i\left(t\right))$ along (12), we obtain

$$\begin{array}{cc}\hfill & D^{+}{V}_i\left(t,z_i\left(t\right)\right)=\lambda exp\left(\lambda (t-t_0)\right){\left|z_i\left(t\right)\right|}^2+exp\left(\lambda \left(t-t_0\right)\right)\mathrm{Re}\left(\overline{z_i\left(t\right)}{\dot{z}}_i\left(t\right)\right)\hfill \\ \hfill =& \lambda exp\left(\lambda (t-t_0)\right){\left|z_i\left(t\right)\right|}^2+exp\left(\lambda \left(t-t_0\right)\right)\mathrm{Re}\left\{\overline{z_i\left(t\right)}\left[\sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)(-d_i^q\left(z_i\left(t\right)\right)\right.\right.\hfill \\ \hfill & +\sum _{j=1}^n(a_{ij}^q{f}_j^q\left(z_j\left(t\right)\right)+b_{ij}^q{f}_j^q\left(z_j\left(t-{\tau }_{ij}^q\left(t\right)\right)\right)+c_{ij}^q{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)f_j^q\left(z_j\left(s\right)\right)\mathrm{d}s)\left]\right\}\hfill \\ \hfill \le & \lambda exp\left(\lambda (t-t_0)\right){\left|z_i\left(t\right)\right|}^2+exp\left(\lambda \left(t-t_0\right)\right)\left\{\left[\sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)(-{\mathcal{D}}_i^q{\left|z_i\left(t\right)\right|}^2\right.\right.\hfill \\ \hfill & +|\overline{z_i\left(t\right)}|\sum _{j=1}^n{\mathcal{L}}_j^q(|a_{ij}^q|\left|z_j\left(t\right)\right|+|b_{ij}^q|\left|z_j\left(t-{\tau }_{ij}^q\right)\right|+|c_{ij}^q|{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)\left|z_j\left(s\right)\right|\mathrm{d}s)\left]\right\}\hfill \\ \hfill \le & exp\left(\frac{\lambda }{2}(t-t_0)\right)\left|z_i\left(t\right)\right|\sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)\left[exp\left(\frac{\lambda }{2}(t-t_0)\right)\left(\frac{\lambda }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right)\left|z_i\left(t\right)\right|\right.\hfill \\ \hfill & +exp\left(\frac{\lambda }{2}(t-t_0)\right)\sum _{j=1}^n{\mathcal{L}}_j^q(|{\tilde{a}}_{ij}^q|\left|z_j\left(t\right)\right|+|{\tilde{b}}_{ij}^q|\left|z_j\left(t-{\tau }_{ij}^q\right)\right|+|{\tilde{c}}_{ij}^q|{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)\left|z_j\left(s\right)\right|\mathrm{d}s)]\hfill \\ \hfill \le & \sqrt{V_i(t,z_i\left(t\right))}\sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)\left[\left(\frac{\lambda }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right)\sqrt{V_i(t,z_i\left(t\right))}+\sum _{j=1}^n{\mathcal{L}}_j^q(|{\tilde{a}}_{ij}^q|\sqrt{V_j(t,z_j\left(t\right))}\right.\hfill \\ \hfill & \left.+exp\left(\frac{\lambda }{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|\sqrt{V_j(t-{\tau }_{ij}^q,z_j(t-{\tau }_{ij}^q))}+|{\tilde{c}}_{ij}^q|{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)exp\left(\frac{\lambda }{2}(t-s)\right)\sqrt{V_j(s,z_j\left(s\right))}\mathrm{d}s)\right],\hfill \end{array}$$

(15)

where $i=1,2,\cdots ,n$. Let $U_i(t,z_i\left(t\right))=\sqrt{V_i(t,z_i\left(t\right))}$. If there is no ambiguity, we indicate $U_i\left(t\right)=U_i(t,z_i\left(t\right))$ for convenience. Substituting them into inequalities (15), we obtain

$$\begin{array}{cc}\hfill D^{+}{U}_i\left(t\right)& \le \frac{1}{2}\sum _{q=1}^m{\varsigma }_q\left(t\right)h_i^q\left(z_i\left(t\right)\right)\left[\left(\frac{\lambda }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right)U_i\left(t\right)+\sum _{j=1}^n{\mathcal{L}}_j^q(|{\tilde{a}}_{ij}^q|U_j\left(t\right)\right.\hfill \\ \hfill & \left.+exp\left(\frac{\lambda }{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|{sup}_{t-{\tau }^q\le s\le t}{U}_j\left(s\right)+|{\tilde{c}}_{ij}^q|{\int }_{-\infty }^t{\chi }_{ij}^q\left(t-s\right)exp\left(\frac{\lambda }{2}(t-s)\right)U_j\left(s\right)\mathrm{d}s)\right],\hfill \end{array}$$

(16)

for $i=1,2,\cdots ,n$.

Defining a curve $\ell =\left\{\mathit{x}\left(l\right):x_i={\xi }_{i}l,l>0,i=1,2,\cdots ,n\right\}$ over ${\mathbb{R}}^n$ and a set $\Lambda \left(\mathit{x}\right)=\left\{\mathit{u}:\mathbf{0}\le \mathit{u}\le \mathit{x},\right.$ $\left.\mathit{x}\in \ell \right\}$. Let $l_0={∥\mathit{\psi }∥}_{t_0}/{\xi }_{min}$, where ${∥\mathit{\psi }∥}_{t_0}={sup}_{s\in \left(-\infty ,t_0\right]}∥\mathit{\psi }\left(s\right)∥$ and ${\xi }_{min}={min}_{1\le i\le n}\left\{{\xi }_i\right\}$. Evidently, $\Lambda \left(\mathit{x}\left(l^{\prime }\right)\right)\subset \Lambda \left(\mathit{x}\left(l^{″}\right)\right)$ as $l^{″}>l^{\prime }$. Defining the set

$$\mathfrak{U}=\left\{\mathit{U}:U_i=exp\left(\frac{\lambda }{2}\left(s-t_0\right)\right)\left|{\psi }_i\left(s\right)\right|,i=1,2,\cdots ,n,-\infty <s\le t_0\right\}.$$

Obviously, $\mathfrak{U}\subset \Lambda \left(\mathit{x}\left(l_0\right)\right)$, i.e.,

$$U_i\left(s\right)=exp\left(\frac{\lambda }{2}\left(s-t_0\right)\right)\left|{\psi }_i\left(s\right)\right|<{\xi }_i{l}_0,-\infty <s\le t_0,i=1,2,\cdots ,n.$$

Furthermore, we assert that $U_i\left(t\right)<{\xi }_i{l}_0$ for $t\ge t_0$; here, $i=1,2,\cdots ,n$. In case it is false, there is at least one indicator $p\in \{1,2,\cdots ,n\}$ and the corresponding moment $t^{\prime }>t_0$ such that $U_p\left(t^{\prime }\right)={\xi }_p{l}_0$, $D^{+}{U}_p\left(t^{\prime }\right)\ge 0$ and $U_i\left(t^{\prime }\right)<{\xi }_i{l}_0$ ($i=1,2,\cdots ,n$, $i\ne p$) for $t_0\le t\le t^{\prime }$. Substituting them into (16) and considering (13), we have

$$\begin{array}{cc}\hfill & D^{+}{U}_p\left(t^{\prime }\right)\hfill \\ \hfill \le & \frac{1}{2}\sum _{q=1}^m{\varsigma }_q\left(t^{\prime }\right)h_p^q\left(z_p\left(t^{\prime }\right)\right)\left[\left(\frac{\lambda }{{\mathcal{H}}_p^q}-{\mathcal{D}}_p^q\right)U_p\left(t^{\prime }\right)+\sum _{j=1}^n{\mathcal{L}}_j^q(|{\tilde{a}}_{pj}^q|U_j\left(t^{\prime }\right)\right.\hfill \\ \hfill & \left.+exp\left(\frac{\lambda }{2}{\tau }^q\right)|{\tilde{b}}_{pj}^q|{sup}_{t^{\prime }-{\tau }^q\le s\le t^{\prime }}{U}_j\left(s\right)+|{\tilde{c}}_{pj}^q|{\int }_{-\infty }^{t^{\prime }}{\chi }_{pj}^q\left(t^{\prime }-s\right)exp\left(\frac{\lambda }{2}(t^{\prime }-s)\right)U_j\left(s\right)\mathrm{d}s)\right]\hfill \\ \hfill \le & \frac{1}{2}\sum _{q=1}^m{\varsigma }_q\left(t^{\prime }\right)h_p^q\left(z_p\left(t^{\prime }\right)\right)\left[\left(\frac{\lambda }{{\mathcal{H}}_p^q}-{\mathcal{D}}_p^q\right){\xi }_p{l}_0+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j{l}_0\left(|{\tilde{a}}_{pj}^q|+exp\left(\frac{\lambda }{2}{\tau }^q\right)|{\tilde{b}}_{pj}^q|+\left|{\tilde{c}}_{pj}^q\right|{{\rm Y}}_{pj}^q\left(\frac{\lambda }{2}\right)\right)\right]\hfill \\ \hfill <& 0.\hfill \end{array}$$

(17)

This conflicts with the statement $D^{+}{U}_p\left(t^{\prime }\right)\ge 0$. Therefore, we obtain $U_i\left(t\right)<{\xi }_i{l}_0$, $i=1,2,\cdots ,n$, i.e., for $t\ge t_0$

$$\left|z_i\left(t\right)\right|<exp(-\frac{\lambda }{2}(t-t_0)){\xi }_i{l}_0=\frac{{\xi }_i}{{\xi }_{min}}{∥\mathit{\psi }∥}_{t_0}exp\left(-\frac{\lambda }{2}(t-t_0)\right),i=1,2,\cdots ,n.$$

Let $M={max}_{1\le i\le n}\left\{{\xi }_i\right\}/{\xi }_{min}$ lead to

$$\parallel \mathit{z}\left(t\right)\parallel <M{∥\mathit{\psi }∥}_{t_0}exp\left(-\frac{\lambda }{2}(t-t_0)\right),t\ge t_0.$$

According to Definition 1, the equilibrium point of (12) demonstrates robust exponential stability, with the exponential convergence rate being $\lambda /2$. □

Let $\mathit{M}$ and $\mathit{N}$ be a non-negative M-matrix and a matrix with non-negative entries, respectively. We refer to ${\nabla }_1\left(\mathit{M}\right)\stackrel{\Delta }{=}\left\{\mathit{\xi }\in {\mathbb{R}}^n|\mathit{M}\mathit{\xi }>\mathbf{0},\mathit{\xi }>\mathbf{0}\right\}$ and ${\nabla }_2\left(\mathit{N}\right)\stackrel{\Delta }{=}$$\left\{\mathit{\xi }\in {\mathbb{R}}^n|\mathit{N}\mathit{\xi }=\rho \left(\mathit{N}\right)\mathit{\xi },\mathit{\xi }>\mathbf{0}\right\}$. In the following, we shall derive criteria that guarantee the robustly exponential stability of (12) using the characteristics of the M-matrix and the result of Lemma 3.

Theorem 2. 

It is supposed that Assumptions 1–3 hold. Provided the following conditions are met:

1. 

${\mathit{D}}_q={\left(d_{ij}^q\right)}_{n\times n}$, $q\in \Xi$ are nonsingular M-matrices, where

$d_{ii}^q={\mathcal{D}}_i^q,i=1,2,\cdots ,n$,

$d_{ij}^q=-{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^q|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right),i,j=1,2,\cdots ,n$ and $i\ne j$;

2. 

$\mathit{D}={\cap }_{q=1}^m{\nabla }_1\left({\mathit{D}}_q\right)$ is nonempty;

then, for ${\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}$, ${\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}$, and ${\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}$, the equilibrium point of (12) not only exists but is also unique, exhibiting robust exponential stability under arbitrary switching. Additionally, the exponential convergence rate λ for (12) can be determined by the inequality (13), corresponding to a given vector $\mathit{\xi }\in \mathit{D}$.

Proof of Theorem 2. 

From condition 1 in Theorem 2, along with Theorem 1, it can be deduced that for any given switching signal, (12) has a unique equilibrium point. Since $\mathit{D}={\cap }_{q=1}^m{\nabla }_1\left({\mathit{D}}_q\right)$ is nonempty, it can be concluded from Lemma 3 that there exists at least one vector $\mathit{\xi }={\left({\xi }_1,{\xi }_2,\cdots ,{\xi }_n\right)}^{\mathrm{T}}\in \mathit{D}\subseteq {\nabla }_1\left({\mathit{D}}_q\right)$ such that

$$-{\xi }_i{\mathcal{D}}_i^q+\sum _{j=1}^n{\xi }_j{\mathcal{L}}_j^q\left(|{\tilde{a}}_{ij}^q|+|{\tilde{b}}_{ij}^q|+|{\tilde{c}}_{ij}^q|\right)<0,$$

(18)

for $i=1,2,\cdots ,n$ and $q\in \Xi$.

Consider functions

$${\mathfrak{F}}_i^q\left(\lambda \right)=\left(\frac{\lambda }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j\left(|{\tilde{a}}_{ij}^q|+exp\left(\frac{\lambda }{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|+\left|{\tilde{c}}_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{\lambda }{2}\right)\right),i=1,2,\cdots ,n;q\in \Xi .$$

(19)

It follows from (18) and (19) that ${\mathfrak{F}}_i^q\left(\lambda \right)\in C^0$ and ${\mathfrak{F}}_i^q\left(0\right)<0$. Calculating the derivative of ${\mathfrak{F}}_i^q\left(\lambda \right)$, we obtain $\mathrm{d}{\mathfrak{F}}_i^q\left(\lambda \right)/\mathrm{d}\lambda >0$. Therefore, there are constants ${\lambda }_i^q>0$, such that

$${\mathfrak{F}}_i^q\left({\lambda }_i^q\right)=\left(\frac{{\lambda }_i^q}{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j\left(|{\tilde{a}}_{ij}^q|+exp\left(\frac{{\lambda }_i^q}{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|+\left|{\tilde{c}}_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{{\lambda }_i^q}{2}\right)\right)=0,$$

(20)

for $i=1,2,\cdots ,n;q\in \Xi$.

Let $0<{\lambda }_{min}<{min}_{1\le i\le n,1\le q\le m}\left\{{\lambda }_i^q\right\}$; then, for $i=1,2,\cdots ,n$ and $q\in \Xi$

$${\mathfrak{F}}_i^q\left({\lambda }_{min}\right)=\left(\frac{{\lambda }_{min}}{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j\left(|{\tilde{a}}_{ij}^q|+exp\left(\frac{{\lambda }_{min}}{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|+\left|{\tilde{c}}_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{{\lambda }_{min}}{2}\right)\right)<0.$$

(21)

With reference to (21) and Lemma 3, it is evident that the unique equilibrium point of (12) is robustly exponentially stable under arbitrary switching. □

Remark 3. 

In Theorem 2, the existence of the positive real number λ in Lemma 3 is proved, which depends on the given vector $\mathit{\xi }\in \mathit{D}$. Therefore, to determine the maximum convergence rate of the system ${\lambda }_{max}$, we can address the optimization problem under the constraint condition ${\mathfrak{F}}_i^q\left(\lambda ,\mathit{\xi }\right)<0$, here $i=1,2,\cdots ,n$, $q\in \Xi$, and $\mathit{\xi }\in \mathit{D}$.

Definition 2. 

Let $\mathcal{IP}=\{t_1,t_2,\cdots \}$ be an impulsive sequence. The number of impulsive times during the interval $\left(t^{\prime },t^{\prime \prime }\right)$ is shown as $\Pi \left(t^{\prime },t^{\prime \prime }\right)$ for any $t^{″}\ge t^{\prime }\ge 0$. If there are constant $T_{\mathcal{IP}}>0$ and $N_{\mathcal{IP}}\ge 0$ such that

$${\Pi }_{\mathcal{IP}}\left(t^{\prime },t^{\prime \prime }\right)\le N_{\mathcal{IP}}+\frac{t^{″}-t^{\prime }}{T_{\mathcal{IP}}},$$

then $T_{\mathcal{IP}}$ is referred to as impulsive average dwell time of $\mathcal{IP}$.

Theorem 3. 

Under Assumptions 1–4, if

1. 

${\mathit{G}}_q={\mathcal{D}}_q-{\mathcal{L}}_q\left(|{\tilde{\mathit{A}}}_q|+|{\tilde{\mathit{B}}}_q|+|{\tilde{\mathit{C}}}_q|\right)$, $q\in \Xi$ are nonsingular M-matrices,

2. 

${\mathit{S}}_q={\cap }_{k=1}^{\infty }\left[{\nabla }_2\left({\mathit{\Phi }}_{1k}\right)\cap {\nabla }_2\left({\mathit{\Phi }}_{2k}\right)\right]\cap {\nabla }_1\left({\mathit{G}}_q\right)$, $q\in \Xi$ are nonempty,

3. 

Impulsive average dwell time $T_{\mathcal{IP}}>T_{\mathcal{IP}}^{\ast }=\frac{ln{\gamma }_{max}}{\beta }$, where $\beta >0$ is determined by inequalities

$$\left(\frac{\beta }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j\left(|{\tilde{a}}_{ij}^q|+exp\left(\frac{\beta }{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|+\left|{\tilde{c}}_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{\beta }{2}\right)\right)<0,$$

for some given vectors $\mathit{\xi }\in {\mathit{S}}_q$ and ${\gamma }_{max}=max{\gamma }_k$, ${\gamma }_k\ge max\left\{1,\rho \left({\mathit{\Phi }}_{1k}\right)+exp\left(\frac{\beta }{2}\tau \right)\rho \left({\mathit{\Phi }}_{2k}\right)\right\}$,

then, for ${\mathit{A}}_q\in {\mathit{A}}_q^{\mathrm{I}}$, ${\mathit{B}}_q\in {\mathit{B}}_q^{\mathrm{I}}$, ${\mathit{C}}_q\in {\mathit{C}}_q^{\mathrm{I}}$, and any switching signal, the equilibrium point of (1a) and (1b) is robustly exponentially stable.

Proof of Theorem 3. 

Since ${\mathit{G}}_q$ are nonsingular M-matrices and sets ${\mathit{S}}_q$ are nonempty (here, $q\in \Xi$), it follows from Theorem 2 that there are at least one vector ${\mathit{\xi }}_{i_0}\in {\mathit{S}}_{i_0}\subseteq {\nabla }_1\left({\mathit{G}}_{i_0}\right)$ and a constant $\beta >0$ such that for $i=1,2,\cdots ,n$ and $q\in \Xi$, inequalities

$$\left(\frac{\beta }{{\mathcal{H}}_i^q}-{\mathcal{D}}_i^q\right){\xi }_i^q+\sum _{j=1}^n{\mathcal{L}}_j^q{\xi }_j^q\left(|{\tilde{a}}_{ij}^q|+exp\left(\frac{\beta }{2}{\tau }^q\right)|{\tilde{b}}_{ij}^q|+\left|{\tilde{c}}_{ij}^q\right|{{\rm Y}}_{ij}^q\left(\frac{\beta }{2}\right)\right)<0$$

(22)

hold. Assuming the $i_p$th subnetwork is activated when $t\in \left(t_p,t_{p+1}\right]$, then we can obtain a switching sequence $\left\{\left(t_0,i_0\right),\cdots ,\right.$$\left.\left(t_p,i_p\right),\cdots |i_p\in \Sigma ,p=0,1,\cdots \right\}$. This, together with the proof of Lemma 3, leads to

$$U_i\left(t\right)<{\xi }_i^{i_0}{l}_0,t\in (-\infty ,t_0]\cup (t_0,t_1],$$

(23)

where $l_0={∥\tilde{\mathit{\phi }}-{\mathit{z}}^{\ast }∥}_{t_0}/{min}_{1\le q\le m,1\le i\le n}\left\{{\xi }_i^q\right\}$.

The following will demonstrate that

$$\left|z_i\left(t\right)\right|<{\gamma }_0{\gamma }_1\cdots {\gamma }_{p-1}{\xi }_i^{\ast }{l}_{0}exp\left(-\frac{\beta }{2}\left(t-t_0\right)\right),i=1,2,\cdots ,n,$$

(24)

for $t_{p-1}<t\le t_p$ ($p=1,2,\cdots$) with the help of mathematical induction, where ${\gamma }_0=1$ and ${\xi }_i^{\ast }={max}_{1\le q\le m}\left\{{\xi }_i^q\right\}$.

When $p=1$, through (23), we know that (24) holds. Now, suppose that for any switching signal and $p=1,2,\cdots ,h$, inequalities

$$\left|z_i\left(t\right)\right|<{\gamma }_0{\gamma }_1\cdots {\gamma }_{p-1}{\xi }_i^{i_{p-1}}{l}_{0}exp(-\frac{\beta }{2}\left(t-t_0\right)),i=1,2,\cdots ,n;t_{p-1}<t\le t_p$$

(25)

hold.

We know from (25) and Assumption 4 that the discrete component of (11a) and (11b) satisfies the following:

$$\begin{array}{cc}\hfill \left|z_i\left(t_h^{+}\right)\right|=& \left|{\varphi }_{1i}^h\left(z_1\left(t_h^{-}\right),\cdots ,z_n\left(t_h^{-}\right)\right)\right|\hfill \\ \hfill & +\left|{\varphi }_{2i}^h\left(z_1\left({\left(t_h-{\tau }_{i1}\left(t_h\right)\right)}^{-}\right),\cdots ,z_n\left({\left(t_h-{\tau }_{in}\left(t_h\right)\right)}^{-}\right)\right)\right|\hfill \\ \hfill \le & \sum _{j=1}^n{\phi }_{ij}^{1h}\left|z_j\left(t_h\right)\right|+\sum _{j=1}^n{\phi }_{ij}^{2h}\left|z_j\left({(t_h-{\tau }_{ij}\left(t_h\right))}^{-}\right)\right|\hfill \\ \hfill \le & \sum _{j=1}^n{\phi }_{ij}^{1h}{\gamma }_0{\gamma }_1\cdots {\gamma }_{h-1}{\xi }_j^{i_{h-1}}{l}_{0}exp\left(-\frac{\beta }{2}\left(t_h-t_0\right)\right)\hfill \\ \hfill & +\sum _{j=1}^n{\phi }_{ij}^{2h}{\gamma }_0{\gamma }_1\cdots {\gamma }_{h-1}{\xi }_j^{i_{h-1}}{l}_{0}exp\left(-\frac{\beta }{2}\left(t_h-{\tau }_{ij}\left(t_h\right)-t_0\right)\right)\hfill \\ \hfill \le & \left(\sum _{j=1}^n{\phi }_{ij}^{1h}{\xi }_j^{i_{h-1}}+exp\left(\frac{\beta }{2}\tau \right)\sum _{j=1}^n{\phi }_{ij}^{2h}{\xi }_j^{i_{h-1}}\right){\gamma }_0{\gamma }_1\cdots {\gamma }_{h-1}{l}_{0}exp\left(-\frac{\beta }{2}\left(t_h-t_0\right)\right),\hfill \end{array}$$

(26)

where $i=1,2,\cdots ,n$. Since ${\mathit{\Phi }}_{1k}$$\left({\mathit{\Phi }}_{2k}\right)$ is non-negative matrix, there is at least one positive eigenvector which corresponds to the non-negative eigenvalue $\rho \left({\mathit{\Phi }}_{1k}\right)$$\left(\rho \left({\mathit{\Phi }}_{2k}\right)\right)$ of ${\mathit{\Phi }}_{1k}$$\left({\mathit{\Phi }}_{2k}\right)$ [35]. By the definition of ${\mathit{S}}_q$, we know that ${\mathit{\xi }}_{i_{h-1}}={\left({\xi }_1^{i_{h-1}},{\xi }_2^{i_{h-1}},\cdots ,{\xi }_n^{i_{h-1}}\right)}^{\mathrm{T}}\in {\nabla }_2\left({\mathit{\Phi }}_{1h}\right)$ and ${\mathit{\xi }}_{i_{h-1}}\in {\nabla }_2\left({\mathit{\Phi }}_{2h}\right)$. Furthermore, we have

$${\mathit{\Phi }}_{1h}{\mathit{\xi }}_{i_{h-1}}=\rho \left({\mathit{\Phi }}_{1h}\right){\mathit{\xi }}_{i_{h-1}},{\mathit{\Phi }}_{2h}{\mathit{\xi }}_{i_{h-1}}=\rho \left({\mathit{\Phi }}_{2h}\right){\mathit{\xi }}_{i_{h-1}},$$

i.e.,

$$\sum _{j=1}^n{\phi }_{ij}^{1h}{\xi }_j^{i_{h-1}}=\rho \left({\mathit{\Phi }}_{1h}\right){\xi }_i^{i_{h-1}},\sum _{j=1}^n{\phi }_{ij}^{2h}{\xi }_j^{i_{h-1}}=\rho \left({\mathit{\Phi }}_{2h}\right){\xi }_i^{i_{h-1}},$$

(27)

where $i=1,2,\cdots ,n$. This, together with ${\gamma }_k\ge max\left\{1,\rho \left({\mathit{\Phi }}_{1k}\right)+exp\left(\frac{\beta }{2}\tau \right)\rho \left({\mathit{\Phi }}_{2k}\right)\right\}$ and (26), leads to

$$\begin{array}{cc}\hfill \left|z_i\left(t_h^{+}\right)\right|\le & \left(\rho \left({\mathit{\Phi }}_{1k}\right)+exp\left(\frac{\beta }{2}\tau \right)\rho \left({\mathit{\Phi }}_{2k}\right)\right){\gamma }_0{\gamma }_1\cdots {\gamma }_{h-1}{l}_0{\xi }_i^{i_{h-1}}exp\left(-\frac{\beta }{2}\left(t_h-t_0\right)\right)\hfill \\ \hfill <& {\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_i^{i_{h-1}}{l}_{0}exp\left(-\frac{\beta }{2}\left(t_h-t_0\right)\right)\hfill \\ \hfill \le & {\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_i^{\ast }{l}_{0}exp\left(-\frac{\beta }{2}\left(t_h-t_0\right)\right),\hfill \end{array}$$

(28)

for $i=1,2,\cdots ,n$.

The following will demonstrate that for $t_h<t\le t_{h+1}$,

$$U_i\left(t\right)<{\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_i^{i_h}{l}_0,i=1,2,\cdots ,n.$$

(29)

In the event that (29) is not true, it implies the existence of certain i and $t^{\prime }\in \left(t_h,t_{h+1}\right]$ such that $D^{+}{U}_i\left(t^{\prime }\right)\ge 0$, and

$$U_i\left(t^{\prime }\right)={\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_i^{i_h}{l}_0,$$

$$U_j\left(t\right)<{\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_j^{i_h}{l}_0\mathrm{for}t<t^{\prime },j=1,2,\cdots ,n.$$

However, by (16) and (21), it is evident that

$$\begin{array}{cc}\hfill & D^{+}{U}_i\left(t^{\prime }\right)\hfill \\ \hfill \le & \frac{1}{2}{h}_i^{i_h}\left(z_i\left(t^{\prime }\right)\right)\left[\left(\frac{\beta }{{\mathcal{H}}_i^{i_h}}-{\mathcal{D}}_i^{i_h}\right){\xi }_i^{i_h}+\sum _{j=1}^n{\mathcal{L}}_j^{i_h}{\xi }_j^{i_h}\left(|{\tilde{a}}_{ij}^{i_h}|+exp\left(\frac{\beta }{2}{\tau }^{i_h}\right)|{\tilde{b}}_{ij}^{i_h}|+\left|{\tilde{c}}_{ij}^{i_h}\right|{{\rm Y}}_{ij}^{i_h}\left(\frac{\beta }{2}\right)\right)\right]{\gamma }_0{\gamma }_1\cdots {\gamma }_h{l}_0\hfill \\ \hfill <& 0.\hfill \end{array}$$

This contradicts the condition $D^{+}{U}_i\left(t^{\prime }\right)\ge 0$. As such, inequalities (29) hold.

That is for $i=1,2,\cdots ,n$ and $t_h<t\le t_{h+1}$,

$$\left|z_i\left(t\right)\right|<{\gamma }_0{\gamma }_1\cdots {\gamma }_h{\xi }_i^{i_h}{l}_{0}exp(-\frac{\beta }{2}(t-t_0)).$$

We may determine that inequalities (24) hold by using mathematical induction. So for $t\ge t_0$

$$\begin{array}{cc}\hfill \left|z_i\left(t\right)\right|<& {\gamma }_0{\gamma }_1\cdots {\gamma }_{k-1}{\xi }_i^{i_{k-1}}{l}_{0}exp(-\frac{\beta }{2}(t-t_0))\hfill \\ \hfill =& exp\left({\sum }_{p=0}^{k-1}ln{\gamma }_p-\frac{\beta }{2}(t-t_0)\right){\xi }_i^{i_{k-1}}{l}_0\hfill \\ \hfill \le & exp\left(\Pi \left(t_0,t\right)ln{\gamma }_{\max }-\frac{\beta }{2}\left(t-t_0\right)\right){\xi }_i^{i_{k-1}}{l}_0\hfill \\ \hfill \le & {\xi }_i^{\ast }{l}_0{\gamma }_{\max }^{N_{\mathcal{IP}}}exp\left(-(\frac{\beta }{2}-\frac{ln{\gamma }_{\max }}{T_{\mathcal{IP}}})\left(t-t_0\right)\right)\hfill \\ \hfill =& \frac{{\xi }_i^{\ast }{\gamma }_{\max }^{N_{\mathcal{IP}}}}{{min}_{1\le q\le m,1\le i\le n}\left\{{\xi }_i^q\right\}}{∥\tilde{\mathit{\phi }}-{\mathit{z}}^{\ast }∥}_{t_0}exp\left(-(\frac{\beta }{2}-\frac{ln{\gamma }_{\max }}{T_{\mathcal{IP}}})\left(t-t_0\right)\right).\hfill \end{array}$$

Let $M=\frac{{\xi }_i^{\ast }{\gamma }_{\max }^{N_{\mathcal{IP}}}}{{min}_{1\le q\le m,1\le i\le n}\left\{{\xi }_i^q\right\}}$ and $\epsilon =\frac{\beta }{2}-\frac{ln{\gamma }_{\max }}{T_{\mathcal{IP}}}$, then we can obtain

$$\parallel \tilde{\mathit{z}}\left(t\right)-{\mathit{z}}^{\ast }\parallel <M{sup}_{s\in \left(-\infty ,t_0\right]}∥\tilde{\mathit{\phi }}\left(s\right)-{\mathit{z}}^{\ast }∥exp\left(-\epsilon \left(t-t_0\right)\right).$$

When $T_{\mathcal{IP}}>T_{\mathcal{IP}}^{\ast }=\frac{ln{\gamma }_{max}}{\beta }$, $\epsilon >0$. In line with Definition 1, the equilibrium point of (1a) and (1b) is robustly exponentially stable, and the exponential convergence rate is $\frac{\beta }{2}-\frac{ln{\gamma }_{\max }}{T_{\mathcal{IP}}}$. □

Remark 4. 

The SCVNN model in this paper, represented by Equations (1a) and (1b), is more general and encompasses several related existing models. For example, ref. [36] investigated the input-to-state stability of a class of switched impulsive memristive neural networks with time-varying delay and stochastic disturbance over $\mathbb{R}$, where only the interconnected matrices within these networks contained switched signals. Additionally, in the field $\mathbb{R}$ of real numbers, ref. [19] studied the stability of impulsive switched neural networks with multiple time delays. Furthermore, refs. [23,28] examined the exponential stability of a class of CVNN with variable and continuously distributed delays of the system state and impulsive effects, without considering switching signals and impulsive delays in the networks.

Remark 5. 

Compared to [31], this paper employs novel analytical methods and introduces a new vector Lyapunov function. These innovations result in equilibrium existence conditions and stability conditions with lower conservatism. Additionally, the model (1) in [31] can be viewed as a special case of the SCVNN model presented in this paper when ${\tilde{h}}_i^{\sigma \left(t\right)}\left({\tilde{z}}_i\left(t\right)\right)=1$, ${\tilde{d}}_i^{\sigma \left(t\right)}\left({\tilde{z}}_i\left(t\right)\right)=e_i^{\sigma \left(t\right)}{\tilde{z}}_i\left(t\right)$, and ${\mathit{B}}_{\sigma \left(t\right)}=\mathbf{0}$.

4. Model Design and Empirical Analysis of PCES

In order to demonstrate the usefulness of the sufficient conditions set in the previous part, a PCES example with simulation results is provided in this section.

First of all, the index system should be determined, which embodies the scientific and comprehensive principle, and the index system determines the network structure of PCES. The indicators of psychological problem characterization shown in

Table 1. Characterization of psychological problems.

Level 1 Indicators	Level 2 Indicators
Cognitive Functions	Memory skills
	Attention
	Comprehension
	Expression
Emotional Responses	Anxiety depression
	Irritability
	Mental instability
	Maladjustment
Interpersonal Communication	Interpersonal alienation
	Interpersonal conflict
	Intimacy
	Obsessive-compulsive behavior
Behavior	Sleepiness
	Crying
	Self-injury/suicide
	Aggressive behavior
Physical Reactions	Dizziness and headaches
	Muscle reactions
	Memory skills

are listed after the psychological state has been analyzed. The development of psychological counseling is then examined, and the indicators of factors influencing the effectiveness of psychological counseling are then listed in

Table 2. Factors influencing the effectiveness of counseling.

Level 1 Indicators	Level 2 Indicators
The Factors of the Client	Motivation for change
	Personality factors
	Reflection and Self-reflection
	Duration of psychological
	Severity of psychological
The Factors of Counselors	Competencies for counselors
	Theoretical techniques of selected counseling
	Counselor’s personality traits
	Degree of compatibility with visitors
Relationship between Consultants and Client	Safe
	Holding
	Sincerely open
Consultation Ssettings	Counseling environment
	Counseling Frequency
	Duration
	Ethical setting
Other factors	Social support systems
	Placebo effect
	Unknown factors

. The two indicator systems’ complete correlation intuitively illustrates the link between the characteristics of psychological problems and the variables affecting how effective psychological counseling is. It is feasible to ascertain the weights between the secondary indicators in Table 1 and Table 2 and combine them to produce connection weight matrices for the key indicators after examining the client’s first evaluation and a number of consultations. The connection weight matrices for the various states will be determined by the evaluation of the various visitation phases. Then, the parameters of each subnetwork can be obtained.

Empirical Analysis: The work process in counseling may be simply split into three phases: initial interview, problem-solving, and end-of-counseling. By weighing the two phases of initial interview and problem solution, we can obtain two subnetworks at different stages. Therefore, consider a PCES with five evaluation indexes and two subnetworks, and the connection weight matrices of each index are given as follows:

Amplification function $h_i^q\left(z_i\left(t\right)\right)=2+sin\left(\right|z_i\left(t\right)\left|\right)$, self-feedback function $d_1^1\left(z_1\left(t\right)\right)=2.1z_1\left(t\right)$, $d_2^1\left(z_2\left(t\right)\right)=2.2z_2\left(t\right)$, $d_3^1\left(z_3\left(t\right)\right)=1.9z_3\left(t\right)$, $d_4^1\left(z_4\left(t\right)\right)=1.8z_4\left(t\right)$, $d_5^1\left(z_5\left(t\right)\right)=2.2z_5\left(t\right)$, $d_1^2\left(z_1\left(t\right)\right)=2.3z_1\left(t\right)$, $d_2^2\left(z_2\left(t\right)\right)=2.2z_2\left(t\right)$, $d_3^2\left(z_3\left(t\right)\right)=1.9z_3\left(t\right)$, $d_4^2\left(z_4\left(t\right)\right)=1.6z_4\left(t\right)$, $d_5^2\left(z_5\left(t\right)\right)=2.6z_5\left(t\right)$, kernel functions ${\chi }_{ij}^q\left(s\right)=exp(-s)$, $s\in [0,+\infty )$, activation functions $f_j^q\left(z_j\left(t\right)\right)=\frac{1}{3}\left(|x_j|+|y_j|\mathrm{i}\right)$, where $z_j=x_j+y_j\mathrm{i}$, $i,j=1,\cdots ,5,q=1,2$.

$$|{\mathit{A}}_1^{\mathrm{I}}|=\left[\begin{array}{ccccc}\left[0.8,1.0\right]& \left[0.5,0.7\right]& \left[0.6,0.8\right]& \left[0.2,0.4\right]& \left[0.1,0.3\right]\\ \left[0.4,0.6\right]& \left[0.0,0.2\right]& \left[0.3,0.5\right]& \left[0.2,0.4\right]& \left[0.6,0.8\right]\\ \left[0.8,1.0\right]& \left[0.3,0.5\right]& \left[0.6,0.8\right]& \left[0.2,0.4\right]& \left[0.1,0.3\right]\\ \left[0.1,0.3\right]& \left[0.6,0.8\right]& \left[1.0,1.2\right]& \left[0.6,0.8\right]& \left[0.2,0.4\right]\\ \left[0.6,0.8\right]& \left[0.4,0.6\right]& \left[1.1,1.3\right]& \left[0.0,0.2\right]& \left[0.3,0.5\right]\end{array}\right]$$

$$|{\mathit{A}}_2^{\mathrm{I}}|=\left[\begin{array}{ccccc}\left[1.0,1.2\right]& \left[0.5,0.7\right]& \left[0.4,0.6\right]& \left[0.4,0.6\right]& \left[0.2,0.4\right]\\ \left[0.5,0.7\right]& \left[0.3,0.5\right]& \left[0.8,1.0\right]& \left[0.1,0.3\right]& \left[0.4,0.6\right]\\ \left[0.2,0.4\right]& \left[1.1,1.3\right]& \left[0.5,0.7\right]& \left[0.0,0.2\right]& \left[0.3,0.5\right]\\ \left[0.3,0.5\right]& \left[0.9,1.1\right]& \left[0.7,0.9\right]& \left[0.1,0.3\right]& \left[0.4,0.6\right]\\ \left[0.2,0.4\right]& \left[0.0,0.2\right]& \left[0.6,0.8\right]& \left[0.7,0.9\right]& \left[0.8,1.0\right]\end{array}\right]$$

$$|{\mathit{B}}_1^{\mathrm{I}}|=|{\mathit{B}}_2^{\mathrm{I}}|=\mathbf{0}$$

$$|{\mathit{C}}_1^{\mathrm{I}}|=\left[\begin{array}{ccccc}\left[0.2,0.4\right]& \left[0.9,1.1\right]& \left[1.1,1.3\right]& \left[0.0,0.2\right]& \left[0.1,0.3\right]\\ \left[1.0,1.2\right]& \left[0.0,0.2\right]& \left[0.5,0.7\right]& \left[0.3,0.5\right]& \left[0.2,0.4\right]\\ \left[0.3,0.5\right]& \left[1.1,1.3\right]& \left[1.1,1.3\right]& \left[0.0,0.2\right]& \left[0.1,0.3\right]\\ \left[0.1,0.3\right]& \left[0.9,1.0\right]& \left[0.7,0.9\right]& \left[1.0,1.2\right]& \left[0.6,0.8\right]\\ \left[0.5,0.7\right]& \left[0.4,0.6\right]& \left[0.6,0.8\right]& \left[0.2,0.4\right]& \left[0.4,0.6\right]\end{array}\right]$$

$$|{\mathit{C}}_2^{\mathrm{I}}|=\left[\begin{array}{ccccc}\left[0.5,0.7\right]& \left[0.6,0.8\right]& \left[1.0,1.2\right]& \left[0.1,0.3\right]& \left[0.3,0.5\right]\\ \left[0.3,0.5\right]& \left[0.2,0.4\right]& \left[0.6,0.8\right]& \left[0.4,0.6\right]& \left[0.4,0.6\right]\\ \left[0.6,0.8\right]& \left[0.6,0.8\right]& \left[0.7,0.9\right]& \left[0.5,0.7\right]& \left[0.2,0.4\right]\\ \left[0.2,0.4\right]& \left[0.5,0.7\right]& \left[0.4,0.6\right]& \left[0.6,0.8\right]& \left[0.4,0.6\right]\\ \left[1.2,1.4\right]& \left[0.5,0.7\right]& \left[1.1,1.3\right]& \left[0.4,0.6\right]& \left[0.6,0.8\right]\end{array}\right]$$

Simple calculating yields ${\mathcal{L}}_q=\mathrm{diag}\left(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$, ${\mathcal{H}}_i^q=1$, where $i=1,\cdots ,5$, $q=1,2$; ${\mathcal{D}}_1=\mathrm{diag}\left(2.1,2.2,1.9,1.8,2.2\right)$, ${\mathcal{D}}_2=\mathrm{diag}\left(2.3.,2.2,1.9,1.6,2.6\right)$, and

$${\mathit{G}}_1=\left[\begin{array}{ccccc}2.0& -0.6& -0.7& -0.2& -0.2\\ -0.6& 2.6& -0.4& -0.3& -0.4\\ -0.5& -0.6& 1.6& -0.2& -0.2\\ -0.2& -0.6& -0.7& 1.2& -0.4\\ -0.5& -0.4& -0.7& -0.2& 2.2\end{array}\right],{\mathit{G}}_2=\left[\begin{array}{ccccc}2.3& -0.5& -0.6& -0.3& -0.3\\ -0.4& 2.4& -0.6& -0.3& -0.4\\ -0.4& -0.7& 1.6& -0.3& -0.3\\ -0.3& -0.6& -0.5& 1.0& -0.4\\ -0.6& -0.3& -0.7& -0.5& 2.5\end{array}\right].$$

Obviously, ${\mathit{D}}_1$ and ${\mathit{D}}_2$ are M-matrices. Let $\mathit{\xi }={\left(1.1570,1.0000,1.3434,2.0960,1.1930\right)}^{\mathrm{T}}\in {\cap }_{q=1}^2$${\nabla }_1\left({\mathit{D}}_q\right)$, so $\mathit{D}$ is nonempty. We may infer from Theorem 2 that the system under consideration has a single equilibrium point that is robustly exponentially stable under any switching.

However, in the actual process of psychological counseling, it is found that the process and effect of psychological counseling will be affected by some unpredictable factors, such as unexpected events in the visitor’s real situation, especially major life events or experiences that will trigger the former traumatic experience, or external objective factors, such as seasonal changes, natural disasters, social hot spots, etc., so that the effect of psychological counseling will show a certain impulse effect. Consider impulsive functions

$${\varphi }_{1i}^k=3.5z_i\left(t_k^{-}\right),{\varphi }_{2i}^k=2.5z_i\left({(t_k-0.5)}^{-}\right),$$

as unpredictable factors of psychological counseling, where $i=1,\cdots ,5,k=1,2,\cdots .$ Obviously, ${\mathit{\Phi }}_{1k}=3.5\mathit{E}$, ${\mathit{\Phi }}_{2k}=2.5\mathit{E}$, and ${\cap }_{k=1}^{\infty }\left[{\nabla }_2\left({\mathit{\Phi }}_{1k}\right)\cap {\nabla }_2\left({\mathit{\Phi }}_{2k}\right)\right]=\left\{\mathit{\xi }\in {\mathbb{R}}^n|{\xi }_i>0,i=1,2,\cdots ,5\right\}$, That is, for $q=1,2$,

$${\mathit{S}}_q={\cap }_{k=1}^{\infty }\left[{\nabla }_2\left({\mathit{\Phi }}_{1k}\right)\cap {\nabla }_2\left({\mathit{\Phi }}_{2k}\right)\right]\cap {\nabla }_1\left({\mathit{D}}_q\right)$$

are nonempty.

Let ${\xi }_1={(1.0000,1.0383,1.1085,1.7874,1.0249)}^{\mathrm{T}}$, ${\xi }_2={(1.0000,1.0181,1.3230,2.0727,1.1734)}^{\mathrm{T}}$, and $\beta =0.99$. Through calculations, we can derive that ${\gamma }_{max}=3.5+exp(\frac{0.9}{2}\ast 0.5)\ast 2.5=6.6308$ and $T_{\mathcal{IP}}^{\ast }=1.9108$. According to Theorem 3, if the impulsive average dwell time exceeds $1.9108$, then the equilibrium point of the investigated system is robustly exponentially stable. In other words, this psychological counseling is effective and stable. Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 present the simulation outcomes for the robust stability of the PCES. The presented results are based on the initial conditions, self-feedback coefficients, and connection matrices outlined in

Table 3. Initial states and interconnection matrices for numerical simulation.

	Subnetwork 1	Subnetwork 2
$\mathit{A}$	$\left[\begin{array}{ccccc}-0.6+0.9i& 0.3+0.6i& -0.4+0.7i& 0.2+0.3i& 0.2+0.2i\\ 0.6-0.2i& -0.2+0.1i& 0.4-0.3i& 0.3-0.3i& 0.7-0.3i\\ 0.5-0.9i& 0.3+0.2i& 0.7+0.4i& -0.1-0.4i& -0.1-0.3i\\ 0.2-0.2i& 0.5-0.6i& 0.8+0.9i& 0.7-0.4i& 0.4-0.1i\\ 0.4-0.7i& -0.4-0.5i& 0.8-i& 0.1-0.2i& 0.4-0.3i\end{array}\right]$	$\left[\begin{array}{ccccc}1.1-0.5i& -0.2+0.7i& 0.5-0.4i& 0.2-0.6i& -0.4+0.2i\\ 0.2+0.7i& 0.4+0.3i& 0.8+0.7i& -0.1-0.3i& 0.2+0.6i\\ 0.2-0.4i& 1.3-0.5i& -0.5-0.5i& -0.1+0.2i& 0.2-0.5i\\ -0.3+0.4i& -0.5-1.0i& 0.5-0.8i& 0.3-0.1i& -0.3+0.5i\\ -0.4-0.2i& -0.2-0.1i& 0.6-0.6i& 0.5-0.8i& 0.8-0.6i\end{array}\right]$
$\mathit{C}$	$\left[\begin{array}{ccccc}-0.3-0.2i& -0.9+0.6i& -0.6+1.2i& -0.2i& 0.2-0.1i\\ 0.5+1.1i& -0.2-0.1i& 0.4-0.6i& -0.4+0.3i& 0.3+0.3i\\ 0.5+0.1i& -0.7-1.1i& 0.2+1.3i& 0.2+0.1i& 0.3+0.1i\\ -0.3-0.1i& 0.8+0.7i& 0.4-0.8i& 0.5+1.1i& 0.7-0.4i\\ -0.7-0.2i& 0.3+0.5i& 0.7-0.4i& -0.4-0.1i& 0.5-0.4i\end{array}\right]$	$\left[\begin{array}{ccccc}0.5-0.5i& -0.4+0.7i& 0.3-1.2i& 0.2+0.2i& -0.5-0.2i\\ 0.3+0.4i& 0.3-0.3i& -0.7+0.4i& -0.6+0.2i& -0.3-0.5i\\ -0.8+0.2i& -0.4+0.7i& 0.2+0.9i& 0.7+0.2i& 0.3-0.2i\\ -0.3-0.3i& 0.5-0.5i& -0.5-0.3i& -0.7+0.4i& -0.5-0.4i\\ 1.3-0.6i& 0.6-0.3i& 1.2-0.6i& 0.5-0.3i& 0.7-0.4i\end{array}\right]$
$\mathit{\phi }\left(s\right)$	${(-0.5\mathrm{i},-1+1.2\mathrm{i},1+1.5\mathrm{i},0.8,-0.7)}^{\mathrm{T}}$	${(0.1+0.5\mathrm{i},1.2,1.5,0.9,0.3)}^{\mathrm{T}}$

.

Theoretical analysis and simulation results indicate that psychological counseling has effective and stable effects on individuals. During counseling, if no external negative stimuli (interference) are present, the counseling effect stabilizes gradually to an acceptable state. If occasional interference occurs during counseling, as long as its intensity does not exceed 2.6 times the original assessment state and no more than one interference happens in two consecutive counseling sessions, the counseling effect also tends to stabilize.

Remark 6. 

Theorems 2 and 3 illustrate the relationship between the robust exponential stability of SCVNN under arbitrary and restricted switching. If ${\mathit{D}}_q$, $q\in \Xi$ consists of nonsingular M-matrices, and SCVNN is at least stable under slow switching. Furthermore, confirming that $\mathit{D}={\cap }_{q=1}^m{\nabla }_1\left({\mathit{D}}_q\right)$ is nonempty guarantees that SCVNN remains stable under arbitrary switching. Since $\mathit{\xi }={\left(1.1570,1.0000,1.3434,2.0960,1.1930\right)}^{\mathrm{T}}\in {\cap }_{q=1}^2{\nabla }_1\left({\mathit{D}}_q\right)$, we may infer from Theorem 2 that the system under consideration has a single equilibrium point that is robustly exponentially stable under any switching. However, the methods from [37,38] only yields stability under restricted switching. In Example 4.1 of [39], setting $X_2=\mathrm{diag}\{0.4,0.4\}$, Theorem 2 of this paper shows that the considered system is robustly exponentially stable under arbitrary switching. In contrast, the method from [39] also only achieves stability under restricted switching. These indicate that the conclusions obtained in this paper are less conservative compared to those in [37,38,39].

Remark 7. 

Manually determining specific parameters is often necessary to achieve desired results when utilizing the LMI technique. In contrast, the sufficient conditions presented in this study are algebraic expressions, making them more practical and intuitive. However, they have the drawback of ignoring the signs of entries in the connection weight matrices. As a result, differences between inhibitory and excitatory effects may therefore go unnoticed.

Remark 8. 

In practical applications of PCES, the primary challenges involve selecting appropriate amplification functions, self-feedback functions, and activation functions. These choices significantly influence the model’s performance and convergence speed. These functions should be selected based on the specific requirements of the task. Different tasks may necessitate different functions to achieve optimal performance.

5. Conclusions

The robust stability of a class of SCVNN with variable and distributed delays, as well as an impulsive influence on the field $\mathbb{C}$ of complex numbers, has been examined in this study. A set of sufficient conditions, employing homeomorphism mapping theory, M-matrix theory, and impulsive average dwell time method, have been formulated to guarantee the existence, uniqueness, and robust exponential stability of the equilibrium point of SCVNN. Based on the analysis of factors related to psychological states and those affecting the effectiveness of psychological counseling, indicators representing psychological problems and factors influencing the effectiveness of psychological counseling have been identified. Furthermore, the correlation between these two types of indicators has been analyzed, leading to the establishment of the PCES. The example of PCES with simulation results is provided to demonstrate the validity and correctness of the presented theories.