Non-fragile mixed H ∞ and passivity control for neural networks with successive time-varying delay components ∗

Department of Mathematics, Thiruvalluvar University, Vellore-632 115, Tamilnadu, India School of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China jdcao@seu.edu.cn Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Neural networks are generally recognized as one of the simplified models of neural processing in the human brain [5].Due to its strong capability of information processing, neural networks have been applied in many areas such as signal and image processing, fault diagnosis, pattern recognition, fixed-point computations, associative memories, the concept of energy, the passivity is the property of dynamic systems and describes the energy flowing through the systems.It relates the input and output with the storage function and thus defines a set of useful input-output properties.The main concept of passivity theory is that the passive properties of a system can keep the system internal stability.The passivity theory is originated from circuit analysis [2] and since then has found successful applications in diverse areas such as stability, signal processing, complexity, chaos control and synchronization, fuzzy control, power control, group coordination, flow control and energy management [3,6,32,44,48,53].Therefore, the problem of passivity analysis for neural networks with time-varying delays have received a great deal of attention and a great many of related literatures have been published [17,30,31,45,49,54].The passivity analysis for switched neural networks with parametric uncertainties have been investigated by Lyapunov theorey and some analysis techniques [11,18].
In recent years, the nonfragile control problem has been an attractive topic in theory analysis and practical implementation.The main topic of the nonfragile control scheme is how to design a feedback control that will be insensitive to some error or gains variation in feedback loop.Therefore, the nonfragile control problem has attracted the interest of many researchers.For example, the problem of nonfragile passivity control for dynamical systems with time-varying delay has been investigated in [10,12,16,23,27,29].The nonfragile passivity and passification problems for a class of nonlinear singular networked control systems with network-induced time-varying delay has been proposed in [16].In [29], the problem of nonfragile H ∞ control has been discussed for memristor-based neural networks using passivity theory.The problem of nonfragile observer-based passive control for a class of Markovian jumping systems subjected to uncertainties and timedelays are investigated in [10].In [23], the authors studied the state estimation problem of H ∞ and passive for memristive neural networks with random gain fluctuations.The nonfragile mixed H ∞ and passive asynchronous state estimation problem for uncertain Markov jump neural networks with time-varying delay is presented in [12].Very recently, the finite-time nonfragile passivity control problem for neural networks with time-varying delay has been studied in [27].However, to the best of authors knowledge, so far, no results on the nonfragile mixed H ∞ and passivity control for neural networks with successive time-varying delay components.This motivates our present research.
Motivated by the above statement, in this paper, we consider the problem of nonfragile mixed H ∞ and passivity control neural networks with successive time-varying delay components.The main contributions of this paper are summarised as follows: • The nonfragile mixed H ∞ and passivity control neural networks with successive time-varying delay components are proposed for the first time.• The required results are derived by using a suitable Lyapunov-Krasovskii function and using linear matrix inequality approach together with Jensen's lemma and Wirtinger-type inequality technique.• Further, the sufficient conditions for the existence of nonfragile state feedback control gain is obtained by using the mixed H ∞ and passivity analysis.• The conditions in our main results can be converted into linear matrix inequalities easily, which can be solved by using Matlab LMI toolbox.
The contributions of the above techniques are demonstrated through two numerical examples.
Notations.Throughout this paper, the superscripts T and −1 denote transpose of a matrix and matrix inverse, respectively; R n and R n×n denote the n-dimensional Euclidean space and set of all n × n real matrices; for symmetric matrices A and B, the notation A > B (respectively A B) means that the matrix A − B is positive definite (respectively nonnegative); symmetric terms in a symmetric matrix are denoted by * ; I is an appropriately dimensioned identity matrix.

Problem formulation and preliminaries
Consider the following neural networks with discrete and distributed time-varying delays: where A is a diagonal matrix; W 0 is the connection weight matrix; W 1 is the discrete delayed connection weight matrix; W 2 is the distributed delayed connection weight matrix; B, B ω and C are known real constant matrices with appropriate dimensions; The time-varying delays σ(t) and ρ(t) satisfied the following conditions: where σ 12 σ 11 , σ 22 σ 21 and µ 1 , µ 2 are constants.Here, let us denote (A1) For any j = 1, 2, . . ., n, there exist constants F − j and F + j such that where For presentation convenience, in the following, we denote We consider the following nonfragile state feedback controller: where K(t) = K + ∆K(t) and K is the controller gain, ∆K is perturbed matrix, which is assumed to be where H a and E a are known real constant matrices with appropriate dimensions, the time-varying matrix F (t) satisfying F T (t)F (t) I.
Definition 1. (See [45].)The neural network ( 1) is said to be asymptotically stable with a mixed H ∞ and passivity performance γ if, under zero initial condition, there exists a scaler γ > 0 such that for all t * > 0 and any nonzero ω(t) ∈ L 2 [0, ∞], where θ ∈ [0, 1] represents a weighting parameter that defines the trade-off between mixed H ∞ and passivity performance.where Lemma 3. (See [47].)Let H, E and F (t) be real matrices of appropriate dimensions with F (t) satisfying F T (t)F (t) I.Then, for any scalar ε > 0,

Main results
In this section, we will propose a sufficient condition of the mixed H ∞ and passivity control for neural networks with successive time-varying delay components and nonfragile controller designs.
Next, we show that the neural network ( 1) is asymptotically stable with a mixed H ∞ and passivity performance γ.To this end, we define the following index: where t * 0.
Corollary 1.Under assumption (A1), for given scalers σ 11 , σ 12 , σ 21 , σ 22 , µ 1 and µ 2 , the neural network (33) is asymptotically stable if there exists positive definite matrices Remark 2. In general, computational complexity will be a big issue based on how large are the LMIs and how more are the decision variables.The results in Theorem 3.1 and Corollary 1 are derived based on the construction of proper Lyapunov?Krasovskii https://www.mii.vu.lt/NA functional with triple and four integral terms and by using Wirtinger-based inequality, Jensen's inequality.It should be mentioned that the derived nonfragile mixed H ∞ passivity criteria for the considered neural networks with time-varying delays is less conservative.Meanwhile, it should also be noticed that the relaxation of the derived results is acquired at the cost of more number of decision variables.As far the results to be efficient enough, it is more comfortable to have larger maximum allowable upper bounds, but still in order to reduce computational burden and time consumption, our future work will be focused on reducing the number of decision variables.
Remark 3. In [1,21,30,37,56], the authors discussed stability, passivity and dissipativity of various models such as Markov jump NNs, BAM NNs, neutral-type NNs, genetic regulatory networks.These models are dealt with only one time-varying delay, but in [25,28,35,41,46,55], the systems are based on a new type of time-varying delay model proposed recently, which contains two time-varying delay components in the state of the dynamical systems, because the system with two additive time-varying delay has a physically powerful application background in a networked control system.Here, it should be mentioned that the passivity control for NNs with the additive time-varying delays.This criterion is derived by defining LKF in (5), which makes full use of the information about σ 1 (t) and σ 2 (t).Therefore, it is of significance to consider nonfragile mixed H ∞ and passivity control problem for neural networks with two additive timevarying delay components.
Remark 4. In [50], the problem of nonfragile robust finite-time H ∞ control for a class of uncertain nonlinear stochastic Itô systems via neural network is addressed.The problem of H ∞ control system with parametric uncertainty in all matrices of the system and output equations have been investigated in [47].In [10], the author addressed the problem of nonfragile observer-based passive control for a class of Markovian jumping systems (MJSs) subjected to uncertainties, nonlinearities and time-delays.In the literature, many control methods have been used such as nonfragile H ∞ control, observer-based passive control and output feedback H ∞ control.However, investigation on nonfragile mixed H ∞ and passivity with successive time-varying delay components have yet to be found in the literature.Motivated by the above discussion, a nonfragile controller mixed H ∞ and passivity for neural networks with successive time-varying delay components, which is different from other existing literature, has been developed in this paper.

Conclusion
The problem of nonfragile mixed H ∞ and passivity control for neural networks with successive time-varying delay components have been presented in this paper.We construct a suitable Lyapunov-Krasovskii function (LKF) with triple and quadruple integral terms and using Wirtinger-type inequality technique.Sufficient conditions are established to ensure the existence of nonfragile mixed H ∞ and passivity analysis.The results are proposed in terms of linear matrix inequalities, which can guarantee the asymptotic stable of the considered neural networks and its nonfragile controller.Finally two examples are presented to illustrate the effectiveness of the proposed criteria.This work can be extended to complex networks and dissipative with Markovian jumping parameter and using delay partitioning approach.This will be done in the near future.

Example 1 .
Consider the neural network (1) with the following parameters:

Figure 1 (
Figure 1(a) denotes the state response of z(t) with the obtained controller gain.Figure 1(b) represents the state response z(t) without controller.It is concluded from Figs. 1(a) and 1(b) that the state trajectories converges to zero quickly, and it demonstrates the efficiency of the proposed controller.
) , U 2 , and positive diagonal matrices H 1 , H 2 , H 3 such that the following LMIs hold: