Improvement of finite-time stability for delayed neural networks via a new Lyapunov-Krasovskii functional

Abstract: The topic of finite-time stability criterion for neural networks with time-varying delays via a new argument Lyapunov-Krasovskii functional (LKF) was proposed and the time-varying delay of the system is without differentiable. For sufficient conditions of this study, a new (LKF) is combined with improved triple integral terms, namely the functionality of finite-time stability, integral inequality, and a positive diagonal matrix without using a free weighting matrix. The improved finite-time sufficient conditions for the neural network with time varying delay are given in terms of linear matrix inequalities (LMIs) and the results show improvement on previous studies. Numerical examples are given to illustrate the effectiveness of the proposed method.


Introduction
Problems of artificial intelligence (AI) can involve complex data or tasks; consequently neural networks (NNs) as in  can be beneficial to overcome the design AI functions manually. Knowledge of NNs has been applied in various fields, including biology, artificial intelligence, static image processing, associative memory, electrical engineering and signal processing. The connectivity of the neurons is biologically weighted. Weighting reflects positive excitatory connections while a negative value inhibits the connection.
Activation functions will determine the outcome of models of learning and depth accuracy in the calculation of the training model which can make or break a large NN. Activation functions are also important in determining the ability of NNs regarding convergence speed and convergence, or in some cases, the activation may prevent convergence in the first place as reported in . NNs are used in processing units and learning algorithms. Time-delay is one of the common distinctive actions in the operation of neurons and plays an important role in causing low levels of efficiency and stability, and may lead to dynamic behavior involving chaos, uncertainty and differences as in . Therefore, NNs with time delay have received considerable attention in many fields, as in .
It is well known that many real processes often depend on delays whereby the current state of affairs depends on previous states. Delays often occur in many control systems, for example, aircraft control systems, biological modeling, chemicals or electrical networks. Time-delay is often the main source of ambivalence and poor performance of a system.
There are two different kinds of time-delay system stability: delay dependent and delay independent. Delayed dependent conditions are often less conservative than independent delays, especially when the delay times are relatively small. The delayed security conditions depend mainly on the highest estimate and the extent of the delay allowed. The delay-dependent stability for interval time-varying delay has been broadly studied and adapted in various research fields in [3, 13-16, 19, 22-24, 28]. Time-delay that varies the interval for which the scope is limited is called interval time-varying delay. Some researchers have reported on NN problems with interval time-varying delay as in [1-5, 7, 11-15, 21, 25], while [16] reported on NN stability with additive time-varying delay.
There are two types of stability over a finite time interval, namely finite-time stability and fixed-time stability. With finite-time stability, the system converges in a certain period for any default, while with fixed-time stability, the convergence time is the same for all defaults within the domain. Both finite-time stability and fixed-time stability have been extensively adapted in many fields such as [26, 29-35, 37, 38]. In [34], J. Puangmalai and et. al. investigated Finite-time stability criteria of linear system with non-differentiable time-varying delay via new integral inequality based on a free-matrix for bounding the integral b aż T (s)Mż(s)ds and obtained the new sufficient conditions for the system in the forms of inequalities and linear matrix inequalities. The finite-time stability criteria of neutral-type neural networks with hybrid time-varying delays was studied by using the definition of finite-time stability, Lyapunov function method and the bounded of inequality techniques, see in [37]. Similarly, in [38], M. Zheng and et. al. studied the finite-time stability and synchronization problems of memristor-based fractional-order fuzzy cellular neural network. By applying the existence and uniqueness of the Filippov solution of the network combined with the Banach fixed point theorem, the definition of finite-time stability of the network and Gronwall-Bellman inequality and designing a simple linear feedback controller.
-To apply to finite-time stability problems of NNs, the time-varying delay is non-differentiable which is different from the time-delay cases in [1-7, 15, 20].
-To illustrate the effectiveness of this research as being much less conservative than the finite-time stability criteria in [1-7, 15, 20] as shown in numerical examples.
To improve the new LKF with its triple integral, consisting of utilizing Jensen's and a new inequality from [34] and the corollary from [39], an action neural function and positive diagonal matrix, without free-weighting matrix variables and with finite-time stability. Some novel sufficient conditions are obtained for the finite-time stability of NNs with time-varying delays in terms of linear matrix inequalities (LMIs). Finally, numerical examples are provided to show the benefit of using the new LKF approach. To the best of our knowledge, to date, there have been no publications involving the problem finite-time exponential stability of NNs.
The rest of the paper is arranged as follows. Section 2 supplies the considered network and suggests some definitions, propositions and lemmas. Section 3 presents the finite-time exponential stability of NNs with time-varying delay via the new LKF method. Two numerical examples with theoretical results and conclusions are provided in Sections 4 and 5, respectively.

Problem formulation
This paper will use the notations as follows: R stands for the sets of real numbers; R n means the n−dimensional space; R m×n is the set of all m × n real matrix; A T and A −1 signify the transpose and the inverse of matrices A, respectively; A is symmetric if A = A T ; If A and B are symmetric matrices, A > B means that A − B is positive definite matrix; I means the properly dimensioned identity matrix. The symmetric term in the matrix is determined by * ; and sym{A} = A + A T ; Block of diagonal matrix is defined by diag{...}.
Let us consider the following neural network with time-varying delays: where x(t) = [x 1 (t), x 2 (t), ..., x n (t)] T denotes the state vector with the n neurons; A = diag{a 1 , a 2 , ..., a n } > 0 is a diagonal matrix; B and C are the known real constant matrices with .., f n (W n x(.))] and g(W(.)) = [g 1 (W 1 x(.)), g 2 (W 2 x(.)), ..., g n (W n x(.))] denote the neural activation functions; is the initial function. The time-varying delay function h(t) satisfies the following conditions: where h 1 , h 2 are the known real constant scalars. The neuron activation functions satisfy the following condition: The neuron activation function f (·) is continuous and bounded which satisfies:
To prove the main results, the following Definition, Proposition, Corollary and Lemmas are useful.

Definition 1.
[34] Given a positive matrix M and positive constants k 1 , k 2 , T f with k 1 < k 2 , the timedelay system described by (2.1) and delay condition as in (2.2) is said to be finite-time stable regarding to (k 1 , k 2 , T f , h 1 , h 2 , M), if the state variables satisfy the following relationship:

Proposition 2.
[34] For any positive definite matrix Q, any differential function z : [bd L , bd U ] → R n . Then, the following inequality holds: Corollary 4.
[39] For a given symmetric matrix Q > 0, any vector ν 0 and matrices J 1 , J 2 , J 3 , J 4 with proper dimensions and any continuously differentiable function z : [bd L , bd U ] → R n , the following inequality holds:

Lemma 6.
[41] For any positive definite symmetric constant matrix Q and scalar τ > 0, such that the following integrals are determined, it has

Main results
Let h 1 , h 2 and α be constants, The notations for some matrices are defined as follows: Let us consider a LKF for stability criterion for network (2.1) as the following equation: Next, we will show that the LKF (3.1) is positive definite as follows: , S > 0 and any matrices P 1 = P T 1 , P 3 = P T 3 , P 6 = P T 6 , P 2 , P 4 , P 5 , such that the following LMI holds: where Remark 8. It is worth noting that most of previous paper [1-7, 15, 20] , the Lyapunov martices P 1 , P 3 and P 6 must be positive definite. In our work, we remove this restriction by utilizing the technique of constructing complicated Lyapunov V 1 (t, x t ), V 3 (t, x t ), V 6 (t, x t ) and V 7 (t, x t ) as shown in the proof of Proposition 7, therefore, P 1 , P 3 and P 6 are only real matrices. We can see that our work are less conservative and more applicable than aforementioned works.

26)
where I ∈ R n×n is an identity matrix, Remark 12. If the delayed NNs as in (2.1) are choosing as B = W 0 , C = W 1 , W = W 2 , then the system turns into the delayed NNs proposed in [23],

27)
where 0 ≤ h(t) ≤ h M andḣ(t) ≤ h D , it follows that (3.27) is the special case of the delayed NNs in (2.1).
and d 2 (t) = 0 and external constant input is equal to zero in Eq (1) of the delayed NNs as had been done in [16], we havė

28)
then (3.28) is the same NNs as in (2.1) that (2.1) is the particular case of the delayed NNs in [16].
Remark 14. If we choose B = 0, C = 1 and g = f and constant input is equal to zero in the delayed NNs in (2.1), then it can be rewritten aṡ
Remark 15. If we set B = W 0 , C = W 1 and W = 1 and constant input is equal to zero in the delayed NNs in (2.1), then (2.1) turns intȯ then (3.30) is the special case of the NNs as in (2.1) which has been done in [8,11,24,28]. Similarly, if we rearrange the matrices in the delayed NNs in (2.1) and set W = 1, it shows that it is the same delayed NNs proposed in [9,19,22].
Remark 16. The time delay in this work is defined as a continuous function serving on to a given interval that the lower and upper bounds for the time-varying delay exist and the time delay function is not necessary to be differentiable. In some proposed researches, the time delay function needs to be differentiable which are reported in [2-6, 8-13, 15-17, 19, 20, 22-24, 28].

Numerical solutions
In this section, we provide numerical examples with their simulations to demonstrate the effectiveness of our results.  Table 1 is to compare the results of this paper with the proposed results in [1-7, 15, 20]. The upper bounds received in this work are larger than the corresponding ones. Note that the symbol '-' represents the upper bounds which are not provided in those literatures and this paper.
As shown in Table 2, the results of the obtained as in [2,3,5,6,20] and this work, by using Matlab LMIs Toolbox, we can summarize that the upper bound of h max is differentiable µ of NNs in (2.1). We can see that the upper bounds received in this paper are larger than the corresponding purposed. Similarly, the symbol '-' represents the upper bounds which are not given in those proposed and this study.

Conclusions
In this research, the finite-time stability criterion for neural networks with time-varying delays were proposed via a new argument based on the Lyapunov-Krasovskii functional (LKF) method was proposed with non-differentiable time-varying delay. The new LKF was improved by including triple integral terms consisting of improved functionality of finite-time stability, including integral inequality and implementing a positive diagonal matrix without a free weighting matrix. The improved finite-time sufficient conditions for the neural network with time varying delay were estimated in terms of linear matrix inequalities (LMIs) and the results were better than reported in previous research.