Less Conservative Stability Criteria for Neutral Type Neural Networks with Mixed Time-Varying Delays

1 School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China 2 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China 3 Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, China 4College of Information Sciences and Technology, Hainan University, Haikou 570228, China


Introduction
During the last two decades, delayed neural networks have drawn a great deal of attention because of their extensive applications in various scientific and technical areas, such as pattern recognition, power systems, parallel computing, signal processing, finance, associative memories, mechanics of structures, and other scientific areas .It is well known that time delay regarded as a major cause of instability and poor performance often appears in many neural networks.Therefore, the stability analysis for delayed neural networks has been investigated extensively in recent few decades.Generally speaking, studying the dynamical behavior of delayed neural networks can be mainly classified into two types: delay-independent stability and delay-dependent stability.As is known to all, delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delay is small.
On the other hand, due to the complicated dynamic properties of the neural cells in the real world, there exist many neural network models such as distributed networks, chemical reactors, and heat exchanges that cannot characterize the properties of a neural reaction process precisely.It is natural and important that these systems will contain some information about the derivative of the past state to further describe and model the dynamics of the complex neural reactions.This new type of neural networks is called neutral neural networks or neural networks of neutral type.However, many researchers have focused on the global stability of neural networks of neutral type only with constant time delay in recent years, which is very restrictive.Hence, described with neutral functional differential equations with discrete and distributed delays, these neural networks called neutral type neural networks with mixed time-varying delays have a lot of research on space.The differential expression not only defines the derivative term of the current state but also explains the derivative term of the past state.Furthermore, it is necessary to have some information about the derivative of the past state in the systems to characterize the dynamics of such complex neural reactions.Practically, neutral type phenomenon always appears in studies of automatic control, chemical reactors, distributed networks, dynamic process including steam and water pipes, population ecology, heat exchanges, microwave oscillators, systems of turbojet engine, lossless transmission lines, vibrating masses attached to an elastic bar, and so on.For this reason, there has been a growing research interest in the study of delayed neural networks of neutral type in the recent years.Therefore, some less conservative stability criteria for neutral type neural networks with mixed time-varying delays have been reported in recently [25,[31][32][33][34][35].Many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as model transformation method, free-weighting matrix method, the method of constructing novel Lyapunov-Krasovskii functionals, delay decomposition technique, and weighting-matrix decomposition method.In [36], the authors derived some less conservative stability criteria by considering some useful terms and using freeweighting matrix technique.By considering the relationship between the time-varying delay and its lower and upper bound, the results obtained in [36] were improved in [37].By constructing a new Lyapunov-Krasovskii functional and using free-weighting matrix method, some more less conservative criteria than those obtained in [37] were proposed in [38].Further, the problems of stability analysis of neutral type neural networks with discrete and distributed delays have been investigated in [39].By using a delay-partitioning approach, a new type of Lyapunov-Krasovskii functionals was constructed to obtain some less conservative stability criteria.However, time delay in [39] is not only constant delay, but also the delay-partitioning approach is equational; hence, this method has some limitations.
Motivated by this technique, it is the first attempt to investigate the integral nonuniform partitioning method to be extended for neutral type neural networks with mixed time-varying delays.In the paper, the reduced conservatism of Theorem 6 benefits from the construction of the new Lyapunov-Krasovskii functionals in (17), which contain some integral nonuniform partitioning method and triple-integral terms, which play an important role in the improvement of less conservative results.Secondly, a novel handling method is given to establish the relationship among ∫  −ℎ ẋ () 5 ẋ (), ∫  −ℎ   () and ( − ℎ), which play an important role in reducing the conservatism of stability criteria further.Furthermore, compared with previous results by using the firstorder convex combination property, our derivation makes full use of the idea of second-order convex combination and the property of quadratic convex function given in the form of a lemma without employing Jensen's inequality.Finally, four numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.Notation 1. Notations used in this paper are fairly standard:   denotes the -dimensional Euclidean space,  × is the set of all  ×  dimensional matrices;  denotes the identity matrix of appropriate dimensions,  stands for matrix transposition, the natation  > 0 (resp.,  ≥ 0), for  ∈  × means that the matrix is real symmetric positive definite (resp., positive semidefinite); diag { 1 ,  2 , . . .,   } denotes block diagonal matrix with diagonal elements   ,  = 1, 2, . . ., , the symbol * represents the elements below the main diagonal of a symmetric matrix, and ⟨⟩  is defined as ⟨⟩  = (1/2)( +   ).

Main Results
In this section we will give sufficient conditions under which the system ( 5) is asymptotically stable.
Similarly, based on Theorem 6, we can obtain the asymptotical stability for system (47) as follows.

Numerical Examples
In this section, four numerical examples are given to show the effectiveness and improvement of the main results proposed in the paper.
where  From Table 1, it can be seen that the stability criterion proposed in the paper is less conservative than those obtained by Feng et al. in [32] and Lakshmanan et al. in [39].Besides, from Tables 1 and 2, the maximum allowable time-delay bounds ℎ will become larger with the values of  becoming larger.Moreover, Figures 1 and 2 show that the state vector () stabilizes to zero asymptotically with different ℎ  and initial values.By using the Matlab LMI toolbox, we solve LMIs ( 13) and ( 14 Example 2. Using this example, we will show the improvement of our results for (52) by two cases (A) and (B).Consider the following parameters as in [32,39]: For the above two Cases A and B of numerical examples, the maximum allowable upper bounds for guaranteeing the asymptotical stability of the corresponding systems obtained from Theorem 13 are listed in Table 3. Table 3   that our results have larger improvement over those results of previous literature.In Case A, when ℎ  = 0, ℎ = , the simulation result is given as Figure 3, which implies that under the given conditions, the state vector () stabilizes to zero asymptotically.
Example 3. Consider the following delayed neural network given in (47) with parameters: The corresponding results for maximum allowable upper bounds of the time-varying delay ℎ() are given in Table 4.
From Table 4, it can be clearly seen that the maximum allowable time-delay bounds ℎ will become small with the values of ℎ  becoming large.Furthermore, when ℎ  = 0.1, the state trajectories of the system (47) are shown in Figure 4.   (69) From Tables 5, 6, and 7, it is clearly shown that Theorem 15 is less conservative than those in the mentioned literature.In Case A, when ℎ  = 0.1, the simulation result is given as Figure 5, which implies that under the given conditions, the state vector () stabilizes to zero asymptotically.Hence, this example indicates fully that the method proposed in the paper plays a major role in reducing conservatism.
Finally, we provide the number of decision variables involved in the LIM in each of the Theorems 6-15 in Table 8.

Conclusions
This paper is concerned with the stability analysis of neutral type neural networks with mixed time-varying delays.Some improved delay-dependent stability results are established by  using a novel approach for the networks.Improved delaydependent stability criteria in terms of linear matrix inequalities (LMIs) are derived by employing a new type of Lyapunov-Krasovskii functionals with three and four integral terms.Different from previous results by using the first-order convex combination property, our derivation applies the idea of second-order convex combination and the property of quadratic convex function.obtained results are formulated in terms of linear matrix inequalities (LMIs).Numerical examples are given to illustrate the effectiveness and the advantage of the proposed main results.

Example 4 .
In this Example, we give four Cases (A)-(C) that show the potential benefits and effectiveness of the developed

Table 2 :
Maximum allowable time-delay bounds ℎ =  for different values ℎ  in Example 1.
) for the case ℎ  = 0 and ℎ = 4.6531 and obtain
clearly shows

Table 4 :
Maximum allowable time delay bounds ℎ for different values ℎ  in Example 3.

Table 5 :
Maximum allowable upper bounds ℎ for different ℎ  in Case A.

Table 6 :
Maximum allowable time-delay upper bounds ℎ in Case B.

Table 7 :
Maximum allowable time-delay bounds ℎ for different values ℎ  in Case C.

Table 8 :
Number of decision variables involved in the stability criteria.