#### Abstract

In this paper, the stability of switched neural networks (SNNs) with interval parameter uncertainties and time delays is investigated. First, the conditions for the existence and uniqueness of the equilibrium point of the system are discussed. Second, the average dwell time approach and M-matrix property are employed to obtain conditions to ensure the globally exponential stability of the delayed SNNs under constrained switching. Third, by resorting to inequality technique and the idea of vector Lyapunov function, sufficient condition to ensure the robust exponential stability of the delayed SNNs under arbitrary switching is derived. The form of the constructed Lyapunov functions is simple, which has certain commonality in studying delayed SNNs, and the proposed results not only are explicit but also reveal the relationship between the constrained switching and the arbitrary switching of the SNNs. Finally, two numerical examples are presented to illustrate the effectiveness and less conservativeness of the main results compared with the existing literature.

#### 1. Introduction

In the past years, neural networks have been widely studied and successfully applied to various realms such as dynamic optimization, associative memory, and pattern recognition and to solve nonlinear algebraic equations and so on [1–5]. In the real world, the connections among different nodes of the networks are not always fixed or consistent, which frequently result in link failure and new link creation. Therefore, the abrupt changes in the structures and parameters of the neural networks often occur, which bring about switchings among certain different topologies and the instability of the networks [6]. In application’s point of view, a fundamental problem of applying neural networks is stability. This is a prerequisite for ensuring that the developed networks can work normally [7–10]. Thus, a popular topic about the stability analysis and stabilization of SNNs has been considered in [11–24].

A switched neural network is a hybrid system, which is essentially composed of a family of subnetworks and a switching signal which defines a specially designated subnetwork being activated at each instant of time. SNNs have attracted significant attention and have been successfully applied to many fields such as artificial intelligence, high-speed signal processing, and gene selection in DNA microarray analysis [25–28]. Generally, a switching system can be described by the following differential equation: where is a family of functions parameterized by some index set and switching signal is a piecewise constant and right continuous function of time mapping from to . The original motivation for the study on switched systems comes partly from that switching among different systems may cause many nonlinear system behaviors such as chaos and multiple limit cycles [29]. In recent years, switched systems have gained increasing attention because many practical systems (for example, constrained robotics, computer-controlled systems, and automated highway systems) can be modeled as switched systems. Furthermore, from the point of view of control, multicontroller switching is an effective way to deal with complex systems. It is well-known that time delays are inevitable in a practical control design which usually leads to unsatisfactory performances and the stability of the dynamic systems may even be destroyed with the increase of delays [30–35]. Attributing to the interaction among the discrete dynamics, continuous dynamics, and time delays, the behaviors of delayed SNNs are very complicated. Besides, due to many inevitable factors such as modelling errors and external perturbations, the models certainly contain uncertainties which can have a serious effect on the dynamical behavior of the systems. To analyze the robustness of the SNNs, one feasible method is to assume that the parameters are included in certain intervals [36]. Therefore, the robust stability analysis of SNNs with interval parameter uncertainties and time delays is of practical and theoretical importance.

For switched dynamical systems, the unpredictable change of system dynamics, such as abrupt perturbation of external environment or sudden change of the system structure due to the failure of a component, may cause the sudden change of the switching signal. In these cases, in order to keep the system working, the system should be stable under arbitrary switching. A typical approach for the stability analysis of switched dynamical systems with arbitrary switching signal is to search for a suitable common Lyapunov function (CLF) such that the rate of the decrease of along the trajectories of systems is not affected by switching (see, e.g., [37–40] and the references therein). If the CLF for the systems does not exist or is not known, in this case, we can study the stability of the system by using multiple Lyapunov functions (MLFs) , (see [37, 41, 42]). However, it is worth noting that to apply this MLF method, one needs to know some information of the state at each switching time. This is to be contrasted with the Lyapunov second method, which do not need to know the knowledge of the solutions. For example, Wu et al. studied the exponential stability of delayed SNNs by using a linear matrix inequality approach and an average dwell time method [12]; based on the piecewise Lyapunov function technique and average dwell time approach, the problem of the exponential stability of SNNs with constant and time-varying delays was investigated, respectively, in [43] and in [44]; by resorting to a novel delay division method, the stability analysis for uncertain SNNs with mixed time-varying delays was addressed. A common feature in these articles is that they all resort to scalar Lyapunov function (or functional). In this paper, the stability of the delayed SNNs with switching signal will be studied by using the idea of vector Lyapunov function with simple forms, which have certain commonality in studying SNNs, and this is the main reason why the obtained results in this paper have less conservativeness. By using the M-matrix property and average dwell time approach, the differential inequalities with time delays will be constructed. By the stability analysis of the differential inequalities, the sufficient conditions to ensure the robust exponential stability of the SNNs under arbitrary switching and constrained switching will be obtained.

Compared with the existing results on SNNs, the contributions of this paper are listed as follows: (a) the forms of the constructed Lyapunov functions are simple, which have certain commonality in studying delayed SNNs under arbitrary switching; (b) unlike asymptotic stability, we analyze the exponential stability of SNNs which include uncertainty and time delays, and the exponential convergence rate can also be obtained; (c) the obtained results not only have less conservativeness but also reveal the relationship between the constrained switching and the arbitrary switching of the delayed SNNs; and (d) comparing with most of the previous results obtained by linear matrix inequalities approach (to apply LMIs approach, one has to determine too many unknown parameters), the proposed criteria are straightforward, which are conducive to practical applications.

*Notation. *Let denote a column vector of (the symbol “” denotes transpose), denote , and denote a vector norm defined by . For , means that each pair of the corresponding elements of and satisfies the inequality “>.” For matrix , denote . denotes the set of continuous functions mapping from to .

#### 2. Preliminaries

The model of a delayed SNNs can be described by the delayed differential equations as follows: where , is the number of neurons, is the state of neuron at time , is the switching signal, which is a piecewise constant and right continuous function of time, and means that the th subnetwork is activated. denotes the neuron self-feedback coefficient matrix of the subnetwork, and represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs; is the activation functions of neurons at time ; and are the connection weight matrices of the th subnetwork, and and denote the connection strengths of the th neuron on the th neuron at time and , respectively; the delay is the bounded function with and ; is the constant external input vector of the th subnetwork. is the initial condition of the system, where .

We assume that the switching signal is unknown a priori. Corresponding to the switching signal , we have a switching sequence , which means that the th subsystem is activated when . We also assume that there is only finite switching in any finite interval and satisfy the following conditions.

*Assumption 1. *Each activation function in the delayed SNNs (2) is assumed to satisfy
for any , where and are known constant scalars and .

*Remark 1. *Assumption 1 was first proposed in [45]. The constants and in this assumption are allowed to be any real number (positive, negative, or zero). Therefore, the activation functions can be nonmonotonic, which are more general than commonly used Lipschitz conditions and sigmoid functions. Such assumption is very useful to obtain less conservative results.

To facilitate the following analysis, let with . In order to study the stability of SNNs under parameter uncertainties, for , the matrices are intervalized as follows:
Define

*Definition 1. *For the delayed SNNs (2), the equilibrium point is said to be robustly exponentially stable if for each , , and , there exist constants and such that
where .

#### 3. Existence and Uniqueness of the Equilibrium Point

The purpose of the present section is to give a sufficient condition which ensures that the equilibrium point of each subsystem satisfies the existence and uniqueness, which implies that for any initial condition , system (2) admits a solution which exists in a maximal interval , where .

*Definition 2. *A real matrix is said to be an M-matrix if , and all successive principal minors of are positive.

Lemma 1 *([46]).**Let be an matrix with nonpositive off-diagonal elements. Then the following statements are equivalent:
*(i)* is an M-matrix*(ii)*There exists a vector such that .*

*Definition 3. *A mapping is a homeomorphism of onto itself if , is one to one, is onto, and the inverse mapping , where denotes the set of continuous functions.

Lemma 2 *([47]).**If satisfies the following conditions:
*(i)* is injective on *(ii)* as **Then is a homeomorphism of .*

Theorem 1. *Under Assumption 1, if for all , are nonsingular M-matrices, then for each specified switching signal , system (2) has a unique equilibrium point.*

*Proof 1. *Because the equilibrium point of subsystems,
satisfies the following equation:
for and . Let
where
for . In the following, we will give a proof that are homeomorphisms of onto itself.

First, we prove that are injective mappings on . Actually, if there exist vectors , , and such that ; then
for and . From Assumption 1, it can be derived that
for . That is,
Let . Obviously, have nonpositive off-diagonal entries and which implies that are nonsingular M-matrices. From Theorem 2.3 of [48], we can get . That is,
which is a contradiction. As a result, are injective mappings on .

Next, we prove that as .

Because are nonsingular M-matrices, we know that there exist positive diagonal matrices , which make matrices positively definite. Let
where
for . Calculate
Using Schwartz inequality, we have
When , we get
which implies as .

Since implies , by Lemma 2, we know that are homeomorphisms of . So each subnetwork has a unique equilibrium point. Therefore, for each specified switching signal , system (2) has a unique equilibrium point. The proof is completed.

#### 4. Exponential Stability of the Delayed SNNs

##### 4.1. Exponential Stability under Constrained Switching

In this section, we will give a sufficient condition ensuring the global exponential stability of delayed SNNs (2) by using the average dwell time method. Let be an M-matrix; we denote

*Definition 4 *(see [37]).

Let denote the number of discontinuities of a switching signal on an interval . is called the average dwell time, if for any and ,
hold.

Theorem 2. *Under Assumption 1, if for all , are nonsingular M-matrices, then for all , , and any external input , the delayed SNN (2) is robustly exponentially stable for any switching signal with the average dwell time satisfying
where is determined by inequalities
for some given and with
*

*Proof 2. *According to Theorem 1, we know that if are M-matrices, then the system has a unique equilibrium point for each specified switching signal. Let be an equilibrium point of system (2) and be any solution of system (2). Denote , , and ; then system (2) can be rewritten as
with .

Due to being M-matrices, by Lemma 1 (ii), we know that there exist and such that
Consider a Lyapunov functional candidate
Calculating the upper right derivative of along the solutions of (25), we get
Defining functions,
Obviously, . Since are continuous functions, there exist , such that . Let ; we can get . Combining it with inequality (28), we get
So for ,
For convenience, we denote when . That is, the th subnetwork is activated for ; then
where
Let ; we can get
Combining (31) and (34) yields
When , the subnetwork is activated; then
where
Combining (27), (35), and (36) yields
Let ; (38) becomes
Let and ; we have
when and . According to Definition 1, equilibrium point system (2) is robustly exponentially stable. The proof is completed.

*Remark 2. *For all , are M-matrices which mean that delayed SNN (2) is globally exponentially stable under constrained switching. From the definitions of functions, we know that the value of relies on vector . So, for obtaining the maximum convergence rate or the minimum average dwell time , one can solve the optimization problem under constraint conditions .

##### 4.2. Exponential Stability under Arbitrary Switching

Define the indicator function where with . Therefore, delayed SNN system (25) can be described as follows:

For any switching signal, only one subnetwork is activated at any time, so it follows that .

Theorem 3. *Under Assumption 1, the equilibrium point of delayed SNNs (2) is robustly exponentially stable for all , , in , and any switching signal if the following conditions are satisfied:
*(i)*, are nonsingular M-matrices*(ii)* is nonempty.**Moreover, the exponential convergence rate of system (2) is equal to , which is determined by
for a given vector .*

*Proof 3. *Consider Lyapunov function candidates . Calculating the upper right derivative of along the solutions of (43), we get