Global Robust Stability of Switched Interval Neural Networks with Discrete and Distributed Time-Varying Delays of Neural Type

By combing the theories of the switched systems and the interval neural networks, the mathematics model of the switched interval neural networks with discrete and distributed timevarying delays of neural type is presented. A set of the interval parameter uncertainty neural networks with discrete and distributed time-varying delays of neural type are used as the individual subsystem, and an arbitrary switching rule is assumed to coordinate the switching between these networks. By applying the augmented Lyapunov-Krasovskii functional approach and linear matrix inequality LMI techniques, a delay-dependent criterion is achieved to ensure to such switched interval neural networks to be globally asymptotically robustly stable in terms of LMIs. The unknown gain matrix is determined by solving this delay-dependent LMIs. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.


Introduction
In the past few decades, neural networks have been a subject of intense research activities due to their wide applications in different areas such as image processing, pattern recognition, associative memory, and combinational optimization.A fundamental problem is the stability which is the prerequisite to ensure that the developed neural network can work 1-40 .In hardware implementation of the neural networks, time delay is inevitably encountered and is usually time varying due to the finite switching speed of amplifiers.It is known that time delay is often the main cause for instability and poor performance of neural networks.Moreover, due to unavoidable factors, such as modeling error, external perturbation, and parameter fluctuation, the neural networks model certainly involves uncertainties such as perturbations and component variations, which will change the stability of neural networks.
To analyze uncertainty of neural networks, one reasonable method is to assume parameters in certain intervals.Therefore, it is of great importance to study the global robust stability of interval neural networks with time-varying delay.Recently, some sufficient conditions for the global robust stability of interval neural networks with time-varying delays and parametric uncertainties have been obtained in terms of LMIs 1-9 .
Since neural networks usually have a spatial extent, there is a distribution of propagation delays over a period of time.In these circumstances, the signal propagation is not instantaneous and cannot be modeled with discrete delays, and a more appropriate way is to incorporate continuously distributed delays in neural network model.On the other hand, in certain physical systems, mathematical models have been described by some functional differential equations of neutral type, which depend on the delays of state and state derivative.In practice, neutral type phenomenon always appear in studies of automatic control, chemical reactors, population ecology, heat exchanges, microwave oscillators, and so on.Hence, the stability for neutral type neural networks with time-varying delay has been also considered in the recent years [10][11][12][13][14][15][16][17][18] .
A class of hybrid systems has attracted significant attention because it can model several practical control problems that involve the integration of supervisory logic-based control schemes and feedback control algorithms.As a special class of hybrid systems, switched systems are regarded as nonlinear systems, which are composed of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching between the subsystems.Recently, switched neural networks, whose individual subsystems are a set of neural networks, have found applications in fields of high-speed signal processing, artificial intelligence, and gene selection in a DNA microarray analysis 19-21 .Therefore, some researchers have studied the stability issues for switched neural networks 22-27 .In 22 , based on the Lyapunov-Krasovskii method and LMI approach, some sufficient conditions were derived for global robust exponential stability of a class of switched Hopfield neural networks with time-varying delay under uncertainty.In 23 , by combining Cohen-Grossberg neural networks with an arbitrary switching rule, the mathematical model of a class of switched Cohen-Grossberg neural networks with mixed time-varying delays were established, and the robust stability for such switched Cohen-Grossberg neural networks was analyzed.In 24 , by employing nonlinear measure and LMI techniques, some new sufficient conditions were obtained to ensure global robust asymptotical stability and global robust stability of the unique equilibrium for a class of switched recurrent neural networks with time-varying delay.In 25 , authors investigated a large class of switched recurrent neural networks with time-varying structured uncertainties and time-varying delay; some delaydependent robust periodicity criteria guaranteeing the existence, uniqueness, and global asymptotic stability of periodic solution for all admissible parametric uncertainties were devised by taking free weighting matrices and LMIs.In 26 , based on multiple Lyapunov functions method and LMI techniques, the authors presented some sufficient conditions in terms of LMIs which guarantee the robust exponential stability for uncertain switched Cohen-Grossberg neural networks with interval time-varying delay and distributed timevarying delay under the switching rule with the average dwell time property.
It should be noted that, in the above literature, almost all results treated of the robust stability for switched neural networks with norm-bounded uncertainty.However, as it well known that there are two forms of parametric uncertainties, namely the interval uncertainty and the norm-bounded uncertainty.To the best of our knowledge, up to now, there are few researchers to deal with the global robust stability for switched neural networks with the interval uncertainty, despite its potential and practical importance.
Motivated by the preceding discussion, the aim of this paper is to present a new class of the switched interval neural networks with discrete and distributed time-varying delays of neural type under interval parameter uncertainties by integrating the theory of switched systems with neural networks.By constructing a suitable Lyapunov-Krasovskii functional and employing Jensens inequality, a delay-dependent criterion will be derived such that the proposed switched interval neural networks are globally robustly asymptotically stable.The proposed criterion is represented in terms of LMIs, which can be solved efficiently by using recently developed convex optimization algorithms 28 .
The rest of this paper is organized as follows.In Section 2, the model formulation and some preliminaries are given.The main result are stated in Section 3. In Section 4, a numerical example is presented to demonstrate the validity of the proposed results.Some conclusions are made in Section 5.
Notations.Throughout this paper, and R denotes the set of real numbers, R n denotes the n-dimensional Euclidean space, R m×n denotes the set of all m × n real matrices.For any matrix A, A T denotes the transpose of A. A > 0 A < 0 means that A is a positive definite negative definite .Given the column vectors x x 1 , . . ., x n T , y y 1 , . . ., y n T ∈ R n , and x T y n i 1 x i y i .ẋ t denotes the derivative of x t , and * represents the symmetric form of matrix.

Neural Network Model and Preliminaries
Consider the interval neural network model with discrete and distributed time-varying delays of neutral type described by the system of differential equations in the form Throughout this paper, the following assumptions are made on the activation functions g j , j 1, 2, . . ., n, the discrete and distributed time-varying delay τ t , and the neutral time-varying delay μ t , where τ N , μ N are constants.The initial value associated with 2.1 is assumed to be x s ψ s , and ψ s is a continuous function on −h, 0 , h max{τ N , μ N }.
With loss of generality, it is assumed that the above neural networks have only one equilibrium point and are denoted by For the purpose of simplicity, the equilibrium x * will be always shifted to the origin by letting y t x t − x * , and the neural network system 2.1 can be represented as follows: where f j y j t g j y j t x * j − g j x * j , and f j 0 0, j 1, 2, . . ., n.The initial condition associated with 2.2 is given in the form y s Based on some transformations, the system 2.2 can be written as an equivalent form

2.3
where where e i ∈ R n denotes the column vector with ith element to be 1 and others to be 0. The switched interval neural networks with discrete and distributed time-varying delays of neural type consists of a set of interval neural network with discrete and distributed time-varying delays of neural type and a switching rule.Each of the interval neural networks is regarded as an individual subsystem.The operation mode of the switched neural networks is determined by the switching rule.According to 2.2 , the switched interval neural network with discrete and distributed delays of neural type can be described as follows: where This means that the matrices A σ t , W 1 σ t , W 2 σ t , W 3 σ t , W 4 σ t are allowed to take values, at an arbitrary time, in the finite set Throughout this paper, it is assumed that the switching rule σ is not known a priori, and its instantaneous value is available in real time.The initial condition associated with the switching system 2.5 is y s ϕ s , s ∈ −h, 0 .By 2.3 , the system 2.5 can be written as where and Δ σ t t satisfies the following quadratic inequality: Define the indicator function ξ t ξ 1 t , ξ 2 t , . . ., ξ N t T , where 1, when the switched system is described by the ith mode, where i 1, 2, . . ., N. Therefore, the system model 2.7 can also be written as where N i 1 ξ i t 1 is satisfied under any switching rules.
To derive the main result in the next section, the following definitions and lemmas will be need.
Definition 2.1.The switched interval neural network model 2.5 is said to be globally robustly asymptotically stable if the neural network model 2.5 is globally asymptotically stable for any Lemma 2.2 see 6 .Let Γ 0 x and Γ 1 x be two arbitrary quadratic forms over R n , then Γ 0 x < 0 for all x ∈ R n − {0} satisfying Γ 1 x ≤ 0 if and only if there exists ε ≥ 0 such that 2.12 Lemma 2.3 Jensen's inequality, see 23 .For any constant matrix Ω ∈ R n×n , Ω Ω T > 0, scalar 0 < γ t < γ, vector function ω : t − γ, t → R n , t ≥ 0 such that the integrations concerned are well defined, then 2.13

Main Results
In this section, the global robust asymptotic stability of the proposed model 2.5 will be discussed.By constructing a suitable Lyapunov functional, a robust delay-dependent criterion for the global asymptotic stability of the neural network system 2.5 is derived in terms of LMIs.
Theorem 3.1.Under the assumptions H 1 and H 2 , if there exist matrices P > 0, . ., 7 such that the following LMIs hold:

3.3
Calculating the time derivative of V t along the trajectory of 2.11 , it can follow that

3.4
By the assumption H 1 and Lemma 2.3, f y s ds .

3.7
Noting the following zero equation with free weighting matrices N i i 1, 2, . . ., 7 , which indicate the relationship between the terms in the state equation 2.11 and can easily be determined by solving the corresponding LMIs, it follows that . By substituting 2.9 and 3.5 -3.8 into 3.4 , it follows that and ε > 0. From Lemma 2.2 with the conditions 3.1 and 3.9 , this implies that V t < 0 for all φ t / 0. Hence, the neural network system 2.7 is globally asymptotically stable for , that is, the switched neural network model 2.5 is globally robustly asymptotically stable.The proof is completed.
When the time-varying delay τ t and μ t in 2.5 becomes into τ t τ const, μ t μ const, according to Theorem 3.1, it is easy to obtain the following corollary.

3.11
then the switched interval neural network model 2.5 is global robust asymptotically stable.
Remark 3.3.When N 1, the switched system model 2.7 degenerated into the interval neural network model 2.1 with discrete and distributed time-varying delays of neutral type which contain neural network models studied in 1-3, 5, 7 .Moreover, note that when N 1, W 1 0, and τ t μ t const, Theorem 3.1 in this paper coincides with Theorem 1 in 12 .Without considering interval parameter uncertainty, the neural network models presented in 11, 13-15 are the special case.Hence, the results obtained in this paper extend and improve the stability results available in the existing literature 1-3, 5, 7, 11, 13-15 .
Remark 3.4.In this paper, augmented Lyapunov functional is used to analyze the stability of the interval neural network model 2.1 with discrete and distributed time-varying delays of neutral type.In the Lyapunov functional, both state and activation function are considered in the same term.Hence, the novel Lyapunov function contains structures more general than the traditional ones, and the negative matrices Π i in Theorem 3.1 contain more elements.This shows that it is easy to find more appropriate elements in Π i to ensure that the LMIs 3.1 hold.Thus, sufficient conditions given in this paper are less conservative than the existing results.
Remark 3.5.In this paper, the activation function is Lipschitz continuous, which is first introduced in 29 and used also in 30, 31 is more general than the usual sigmoid functions.Therefore, the stability results obtained in this paper are less conservative than those in 10-13, 27 .
Remark 3.6.In 32, 33 , the mixed time-delay problems have been considered for the stochastic system with Markovian jump parameters and discrete-time stochastic complex networks with randomly occurred nonlinearities.By applying the Lyapunov-Krasovskii functional approach and linear matrix inequality techniques, the conditions of exponential stabilization and global synchronization were presented for the stochastic system and discrete-time stochastic complex networks.In 34 , Bounded H-infinity synchronization and state estimation were considered for discrete time-varying stochastic complex networks over a finite-horizon via linear matrix inequality.35 The authors studied the robust H-infinity fuzzy output-feedback control; a controller design is given for the system with multiple probabilistic delays and multiple missing measurements.In the future, based on 32, 33 , the model of the switched interval stochastic system and the switched interval discrete-time stochastic complex networks will be expected to be established, and the stability strategy proposed in this paper will be utilized to investigate the stability problems.

An Illustrative Example
In this section, an example will be given to illustrate the validity and effectiveness of the proposed stability criterion for the switched interval neural network with discrete and distributed delays of neural type when N 2.
Example 4.1.Consider the following second-order switched interval neural networks with discrete and distributed delays of neural type:   Let  2 displays the state trajectories of this network with initial value ϕ 1 t , ϕ 2 t T cos 3t 2 − 0.5, sin 2t 2 − 0.6 T , t ∈ −1, 0 .It can be seen that these trajectories asymptotically converge to the unique equilibrium x * 0, 0 T of the network.This is in accordance with the conclusion of Theorem 3.1.

Conclusion
In this paper, a novel class of switched interval neural networks with discrete and distributed delays of neural type has been presented by combing the theories of the switched systems and the interval neural networks with discrete and distributed delays of neural type.Furthermore, a delay-dependent criterion expressed in the form of LMIs has been obtained to guarantee the proposed neural networks to be globally asymptotically robustly stable under interval parameter uncertainties.An illustrative example has been also given to demonstrate the effectiveness of the proposed LMI-based stability criteria.
Solving the LMI in 3.4 by using appropriate LMI solver in the Matlab, the feasible positive definite matrices P, Q i , i 1, 2, 3, 4 and the matrices N i , i 1, 2, . . ., 7 could be as By Theorem 3.1, this switched interval neural network with discrete and distributed delays of neural type is globally robustly asymptotically stable under any switching rules.
A 1 A 1 , W 11 W 11 , W 21 W 21 , W 31 W 31 , W 41 W 41 and A 2 A 2 , W 12 W 12 , W 22 W 22 , W 32 W 32 , and W 42 W 42 .For numerical simulation, assume that the two subsystems are switched every four seconds.Figure 1 displays the state trajectories of this network with initial value ϕ 1 t , ϕ 2 t T cos 2t 2 − 0.5, sin 3t 2 − 0.4 T , t ∈ −1, 0 .It can be seen that these trajectories asymptotically converge to the unique equilibrium x * 0, 0 T of the network.This is in accordance with the conclusion of Theorem 3.1.Let A 1 A 1 , W 11 W 11 , W 21 W 21 , W 31 W 31 , W 41 W 41 and A 2 A 2 , W 12 W 12 , W 22 W 22 , W 32 W 32 , and W 42 W 42 .For numerical simulation, assume that the two subsystems are switched every four seconds.Figure