New Delay-Dependent Exponential Stability Criteria for Neural Networks with Mixed Time-Varying Delays

This study is concerned with the problem of new delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays via introducing a novel integral inequality approach. Specifically, first, by taking fully the relationship between the terms in the Leibniz-Newton formula into account, several improved delay-dependent exponential stability criteria are obtained in terms of linear matrix inequalities (LMIs). Second, together with some effective mathematical techniques and a convex optimization approach, less conservative conditions are derived by constructing an appropriate Lyapunov-Krasovskii functional (LKF). Third, the proposed methods include the least numbers of decision variables while keeping the validity of the obtained results. Finally, three numerical examples with simulations are presented to illustrate the validity and advantages of the theoretical results.


Introduction
Over the course of the past decade, neural networks have become an important area of research and attracted increasing attention due to their extensive applications in many practical systems, such as power systems [1], pattern recognition [2], signal detection [3], landmark recognition [4], and other scientific areas [5][6][7].
Moreover, it is inevitable to introduce time delay into the signals transmitted among neurons because the processes of transcription and translation are not instantaneous.However, it is a well-known fact that time delay as a source of instability and poor performance usually appears in many dynamical systems, for instance, Cohen-Grossberg neural networks, cellular neural networks, BAM neural network, chaotic neural networks,  ∞ filtering, and nonlinear systems .Therefore, stability analysis for neural networks with delays has been an attractive subject of research in recent years [50][51][52][53].
Furthermore, neural networks (NNs) often have a spatial nature due to the presence of many parallel pathways of a variety of axon sizes and lengths.Thus, in order to have a more accurate model, a distributed delay over a certain time of duration needs to be included in NNs such that the distant past has less influence compared to the recent behavior of the state.Therefore, there has been a growing interest in the study of neural networks with discrete and distributed delays during the past two decades.To date, some results on delay-dependent exponential stability for neural networks with mixed time-varying delays have been reported in [18][19][20][21][22][23][34][35][36].In [18], the authors considered the global asymptotic stability for a class of delayed cellular neural networks with mixed time-varying delays by using LMIs approach, Lyapunov theory, and Leibniz-Newton formula.However, the activation functions in [18] were assumed to be monotonically nondecreasing.In [19], several delaydependent sufficient conditions are obtained to guarantee the global asymptotic and exponential stability of the addressed neural networks by employing appropriate LKF and linear matrix inequality (LMI) technique.In [20], an exponential stability criterion is proposed by constructing an augmented LKF, where the discrete delay () must be differentiable.In [22,23], some improved delay-dependent stability criteria are derived in terms of linear matrix inequalities by dividing 2 Mathematical Problems in Engineering the discrete delay interval into multiple segments.Different from [35], the appropriate LKF not only divides the discrete delay interval [0, ] into two ones [0, /2] and [/2, ], but also divides the discrete delay interval [0, ] into three ones [0, ()/2], [()/2, ()], and [(), ].Although this approach seems to be effective for achieving less conservative conditions, it can increase the larger numbers of computed variables.Hence, there exists great room for further improvement.To the best of our knowledge, it is of a great significance for the current research to find a more effective approach to get rid of the strict constraint and obtain less conservative conditions.
Motivated by the above discussion, combining effective mathematical techniques and a convex optimization approach, we choose a more general type of LKF to study the delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays in the paper.Some improved delay-dependent stability conditions derived benefit mainly from using firstly a new integral inequality approach, which is proved to be less conservative than the celebrated Jensen's inequality and showed having a great potential efficient in practice.Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the proposed method.Besides, the main merit of this method lies in containing the least numbers of decision variables while keeping the validity of the obtained results.Finally, the stability criteria obtained turn out to be less conservative than some recently reported ones via three numerical examples.
Notation.Notations used in this paper are fairly standard: R  denotes the -dimensional Euclidean space, and R × is the set of all × dimensional matrices;  is the identity matrix of appropriate dimensions, and   is the matrix transposition of the matrix .By  > 0 (resp.,  ≥ 0), for  ∈ R × , we mean that the matrix  is real symmetric positive definite (resp., positive semidefinite); diag{ 1 ,  2 , . . .,   } denotes block diagonal matrix with diagonal elements   ,  = 1, 2, . . ., , and the symbol * represents the elements below the main diagonal of a symmetric matrix; ⃗  is defined as ⃗  =  +   .
Assumption A. The time-varying delay ℎ() is continuous and differential function satisfying (2) Assumption B. For the constants  −  and  +  the bounded activation function   (⋅) in (1) satisfies the following condition: We denote Under Assumption B, by using Brouwer's fixed-point theorem [25], it can be easily proven that there exists one equilibrium point for system (1).Assuming that  * = [ * 1 , . . .,  *  ]  is an equilibrium point of system (1).For convenience, we firstly shift the equilibrium point  * to the origin by letting () = ()− * and (()) = (())−( * ), and then system (1) can be converted to where (()) = [ 1 ( 1 ),  2 ( 2 ), . . .,   (  )]  .It is easy to check that the function   (⋅) satisfies   (0) = 0, and Due to the influence of external factors, ∫  − (())  cannot express the actual state of the accurate information.Therefore, by translating  to function () (0 ≤ () ≤ ), we have In the paper, we will attempt to formulate some practically computable criteria to check the global exponential stability of system (6).The following lemmas are useful in deriving the criteria.

Main Results
In this section we will give sufficient conditions under which system ( 6) is globally exponentially stable.
Theorem 6.For given scalars  > 0, ℎ > 0, and 1 > ℎ  , the origin system (6) with the neuron activation function (()) satisfying condition (5) and the time-varying delay ℎ() satisfying ( 2) is globally exponentially stable with the exponential convergence rate index  if there exist  > 0, such that the following symmetric linear matrix inequality holds: where Proof.Consider an augmentation of LKF for system (6) as follows: where The time derivative of (  ) along with the trajectory of system ( 6) is given as where Using Lemma 4, we can have Using Lemma 5, we may get ] By Lemma 3, we can obtain From ( 5), for any  ×  diagonal matrices   > 0 ( = 1, 2, . . ., 4), the following inequality holds: Furthermore, for arbitrary matrices  1 ,  2 ,  3 , and  4 with appropriate dimensions, we have The combination of ( 19)- (24) gives where From ( 14), we know that V(  ) < 0, which means the asymptotically stability of system (5).This completes the proof.
Furthermore, setting  = max{ℎ, }, we can have It is easy to have According to 2   ≤    +    −  with  > 0, Thus according to ( 26)-( 28), there exists a positive constant  such that where On the other hand, we have Therefore Then, from Definition 1, system ( 6) is exponentially stable with convergence rate , and the proof is completed.
Remark 7. In the paper, the reduced conservatism of Theorem 6 benefits primarily from a new integral inequality, which is proved to be less conservative than the celebrated Jensen's inequality, and takes fully the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs into account.In order to lower the conservatism of stability criteria, we further deal with the integral terms of −ℎ ż  ()( 3 −  33 ) ż ()  via Lemma 4. Different from that of [17], this kind of processing method can reduce ulteriorly the conservatism of stability criteria.
Remark 8.As a matter of fact, Theorem 6 gives a stability criterion for system (6) with ℎ() satisfying 0 ≤ ℎ() ≤ ℎ, 0 ≤ ḣ () ≤ ℎ  , where ℎ  is given constant.In many cases, ℎ  is unknown.Considering this situation, a rate-independent corollary for the delay ℎ() satisfying 0 ≤ ℎ() ≤ ℎ is derived by setting  2 = 0,  4 = 0, and  2 = 0 in the proof of Theorem 6. Theorem 9.For given scalars 0 <  and ℎ > 0, the origin of system (6) with the neuron activation function (()) satisfying condition ( 5) is globally exponentially stable with the exponential convergence rate index  if there exist  > 0, ] > 0, such that the following symmetric linear matrix inequality holds: where The other procedure is straight forward from the proof of Theorem 6, so we omit it.
Remark 10.In the paper, we make full use of the relationship between ∫ (())  were always ignored in [18][19][20], which may lead to considerable conservatism to certain extent.
Remark 11.Due to constructing a simple type of Lyapunov-Krasovskii functional and taking full advantage of effective mathematical techniques, the conservatism of improved delay-dependent stability criteria obtained is reduced to a great degree in this study.Compared with those in previous articles [22,23], we employ a few free variables and do not use a delay decomposition method and add some zero terms, that is, not only dividing the discrete delay interval [0, ] into two ones [0, /2] and [/2, ], but also dividing the discrete delay interval [0, ] into three ones [0, ()/2], [()/2, ()], and [(), ] and adding the following equalities: By using this method, the conservatism of the obtained stability condition in [22,23] is reduced to some degree.However, the computing complexity is also improved since more variables are involved.Besides, we provided a comparison of the numbers of the variables involved in [22,23] and our paper in Table 1.From Table 1, it is clear to see that the number of decision variables in our paper is much less than those in [22,23].Thus, it also expounds validity and applicability of the proposed method.
Remark 12.In many actual applications, maximum allowable time-delay upper bounds ℎ are of interest.In Theorems 6 and 9, with a fixed ℎ  and ,  can be obtained through following optimization procedure: Maximize ℎ, Subject to (14) or (34) .
Besides, maximum allowable time-delay upper bounds  obtained are very valuable.In Theorems 6 and 9, with a fixed ℎ  and , ℎ can be also acquired through following optimization procedure: Maximize , Subject to (14) or (34) .
Inequalities ( 37) and ( 38) are a convex optimization problem and can be obtained efficiently by using the MATLABLMI Toolbox.

Numerical Examples
In this section, three examples are given to demonstrate the feasibility and effectiveness of the main results derived above.
For different ℎ and ℎ  , the allowable upper bounds of the exponential convergence rate index  calculated by Theorem 6 in this paper and Theorem 1 in [20][21][22][23][24] are listed in Table 2.According to Table 2, this example is given to indicate significant improvements over some existing results.
Besides, for the parameters listed above, let  = 0.2, ℎ = 0.5, ℎ  = 0.5, and  = 0.65.Then we can obtain the following feasible parameters by Theorem 6 in our paper.Due to the limitation of the length of this paper, we only provide a part of the feasible solutions here as follows: Case B (set  − = diag{0, 0, 0},  + = diag{0.5,0.5, 0.5}).
Case A. For ℎ =  the corresponding upper bounds of ℎ for unknown ℎ  obtained by Theorem 9 and the results in [18][19][20] are listed in Table 4.According to Table 4, this example shows that the stability criterion in this paper gives much less conservative results than those in the previous literatures.Case B. For ℎ = 1, the corresponding upper bounds of  for unknown ℎ  obtained by Theorem 9 are listed in Table 5.From Table 5, it is shown clearly that our results have significant improvement over the existing results.

Conclusions
In this paper, the delay-dependent exponential stability problem for NNs with mixed time-varying delays has been investigated.By using a new integral inequality approach to express the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs for the first time, several less conservative delay-dependent exponential stability criteria are obtained.Moreover, combining effective mathematical techniques and a convex optimization approach, new delay-dependent exponential stability conditions are derived by constructing a proper LKF.Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed methods.The foregoing

Example 3 .
Consider a delayed neural network in(6) with parameters as follows:

Figure 6 :
Figure 6: State trajectories of () on the space for ℎ = 5.220 in Example 2.

Table 1 :
Comparison of the numbers of the involved variables.

Table 2 :
Allowable upper bounds of  for  for Case A in Example 1.

Table 3 :
Maximum allowable time delay upper bounds ℎ for Case B in Example 1.

Table 4 :
Maximum allowable time delay upper bounds ℎ =  in Example 2.

Table 5 :
Maximum allowable time delay upper bounds  for different values  in Example 2.

Table 6 :
Comparison of the maximum delay ℎ between different methods for various ℎ  in Example 3.