Finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays

doi:10.1016/j.fss.2019.04.004

Fuzzy Sets and Systems

Volume 381, 15 February 2020, Pages 51-67

https://doi.org/10.1016/j.fss.2019.04.004 Get rights and content

Abstract

This paper focuses on the finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays. To end this, by choosing proper variable transformation, the original system can be rewritten as the first order differential system. Based on finite-time stability theory and combining with inequality techniques and some analysis methods, some novel delay-independent criteria in terms of algebraic inequalities are obtained to ensure that finite-time synchronization can be achieved between the drive system and the response system by designing two different types of controllers. The criteria here are very simple to implement in practice and avoid complex computation on the matrix inequalities. Moreover, the settling time is also estimated. Finally, two numerical examples with simulations are presented to show the effectiveness of the obtained results.

Introduction

In 1986, Babcock and Westervelt [1] first introduced an inertial term into neural networks and pointed out that the dynamics could be complex when the neuron couplings contain an inertial nature, because the addition of inertial terms can generate complicated bifurcation behavior and chaos [2]. Now, the dynamic behaviors of all kinds of inertial neural networks (INNs) have been deeply studied and some interesting results have been obtained in the existence and stability of the equilibrium point [3], [4], [5], [6], [7], [8] and periodic and anti-periodic solutions [9], [10], [11], [12], Lagrange stability [13], dissipativity [14], [15], passivity [16], convergence [17], exponential stabilization [18], long time synchronization [19], [20], [21], [22], [23], [24], [25] and finite-time synchronization (FTS) [26], [27], [28].

It is worth noting that the majority of the above mentioned literatures studied asymptotic behavior of INNs described with first order differential equations by choosing suitable reduced-order variable substitutions. To our knowledge, there is only few paper that investigated the asymptotic stability and synchronization of INNs with delays using nonreduced order method [21]. Meanwhile, it is regrettable that extremely few publications [26], [27], [28] on FTS for INNs were reported. Currently, two controllers were added into the deformed system of the original response system to make the discussed INNs achieve FTS [26], [27] and infinite-time synchronization [24], [25], respectively. It is an obvious fact that two controllers are added to the transformational first-order systems in [24], [25], [26], [27]. Actually, the fewer controllers, the easier it is to implement in practice. So far, there is only few publication [28] that studied the FTS for the concerned INNs by a delay-dependent controller.

In addition, time delays widely exist in various fields such as electronic and biological neural systems. In real world, time delays may be proportional. In fact, the proportional delay function is a monotonically increasing function and also proved to be useful in web services [29]. Its presence leads to an advantage is that the network's running time can be controlled based on the maximum delay allowed by the network [30]. Meanwhile, the changes of the current state of system may be related to not only the current specific state but also the past state and the changes of the past state. For this reason, neutral term is introduced as a kind of more extensive time delay that refers to the time delay in the derivative of the state function. Because of the importance of neutral term, it attracted the attention of scholars [31], [32]. Besides, it is worth noting that non-autonomous phenomena often occur in many realistic systems, for instance, when considering a long-term dynamical behavior of the system, the time-varying parameters of the system usually change along with time and can result in the oscillation of the system [11], [31], [33], [34]. Therefore, the asymptotic behavior analysis for non-autonomous neural networks with delays, especially proportional delays and neutral-type delays is of interest and important. Nowadays, considerable attention has been paid to the dynamics analysis for all kinds of neural networks with various delays [3], [4], [5], [6], [8], [11], [13], [30], [31], [32], [33], [34], [35], [36].

However, besides various delays, in mathematical modeling of practical issues, we can also encounter some other inconveniences, for example, the complexity, the approximation and the vagueness. Fuzzy logic systems have been proved to be universal approximators, i.e., they can approximate any nonlinear functions. In [37], [38], fuzzy cellular neural networks (FCNNs) were first introduced. Recently, various kinds of fuzzy neural networks (FNNs) have been widely adopted for nonlinear systems [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51]. In [39], [40], [41], the authors studied the stability problem of T-S FNNs described by a set of IF-THEN rules. In [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], the researchers investigated asymptotic properties for various FNNs with the fuzzy AND and fuzzy OR operation.

Synchronization between neural networks has been widely used in secure communication, image processing, and so on. The synchronization of the drive-response systems can eventually be translated into the stability analysis of the error systems. In addition, compared with the asymptotic synchronization, the FTS of neural networks is paid more and more attention by researchers because of its usefulness [26], [27], [28], [47], [48], [49], [51]. To the best of our knowledge, the stability analysis of neutral-type inertial neural networks (NTINNs) with delays has been investigated in [32], the global synchronization [19], [20], [21], [22], [23], [24], [25] and the FTS [26], [27], [28] for various kinds of INNs have been obtained, respectively. But there is hardly any paper that considered the FTS for fuzzy neutral-type inertial neural networks (FNTINNs) with time-varying coefficients and proportional delays. These constitute the motivation for the present research.

Motivated by the above analysis, we will attempt to integrate the fuzzy AND and fuzzy OR operation into INNs to express a class of NTINNs with time-varying coefficients and proportional delays. The main purpose of this paper is to study the FTS of FNTINNs with proportional delays. The main contributions of this paper are the following aspects: (i) The present paper is one of the first papers that attempts to study the FTS of FNTINNs with time-varying coefficients and proportional delays. (ii) Two control techniques are adopted to achieve the FTS, including designing two controllers and only one controller. (iii) The established sufficient conditions are expressed in terms of algebraic inequalities, which are very simple to implement in practice and avoid complex computation on the matrix inequalities.

The rest of this paper is organized as follows. Section 2 introduces the FNTINNs and their drive and response systems and presents some preliminaries including some necessary definitions and lemmas. The main results are obtained in Section 3. Section 4 gives two numerical examples with simulations to confirm the validity of our results. Finally, conclusions are presented in Section 5.

Section snippets

Preliminaries

For convenience, let $R$ and $R^{+}$ represent the set of all real and positive real numbers, respectively. For a bounded and continuous function $S (t)$ , note $S^{+} = \sup_{t \in R} S (t)$ , $S^{-} = \inf_{t \in R} S (t)$ .

Consider the following FNTINNs with proportional delays for $i \in I = {1, 2, \dots, n}$ $\frac{d^{2} x_{i} (t)}{d t^{2}} = - a_{i} (t) \frac{d x_{i} (t)}{d t} - b_{i} (t) x_{i} (t) + F_{i} (x (t)),$ where $\begin{matrix} F_{i} (x (t)) = & \sum_{j = 1}^{n} c_{i j} (t) f_{j} (x_{j} (t)) + \sum_{j = 1}^{n} d_{i j} (t) f_{j} (x_{j} (q_{i j} t)) + \sum_{j = 1}^{n} f_{i j} (t) f_{j} ({\dot{x}}_{j} (q_{i j} t)) + \sum_{j = 1}^{n} g_{i j} (t) ν_{j} (t) \\ + ⋀_{j = 1}^{n} T_{i j} (t) ν_{j} (t) + ⋁_{j = 1}^{n} H_{i j} (t) ν_{j} (t) + ⋀_{j = 1}^{n} h_{i j} (t) f_{j} (x_{j} (q_{i j} t)) + ⋁_{j = 1}^{n} k_{i j} (t) f_{j} (x_{j} (q_{i j} t)) \\ + ⋀_{j = 1}^{n} m_{i j} (t) f_{j} (\dot{x} \end{matrix}$

Main results

In this section, we consider the FTS of FNTINNs with different control inputs. The control inputs are designed as follows: ${\begin{matrix} {\bar{u}}_{i} (t) = & - {\bar{λ}}_{i} e_{i} (t) - η_{1} sign (e_{i} (t)) | e_{i} (t) |^{θ}, \\ {\tilde{u}}_{i} (t) = & - {\tilde{λ}}_{i} z_{i} (t) - k_{i} \sum_{j = 1}^{n} d_{i j} (t) f_{j} (e_{j} (q_{i j} t)) - k_{i} \sum_{j = 1}^{n} f_{i j} (t) f_{j} ({\dot{e}}_{j} (q_{i j} t)) \\ - k_{i} ⋀_{j = 1}^{n} h_{i j} (t) f_{j} (e_{j} (q_{i j} t)) - k_{i} ⋁_{j = 1}^{n} k_{i j} (t) f_{j} (e_{j} (q_{i j} t)) - k_{i} ⋀_{j = 1}^{n} m_{i j} (t) f_{j} ({\dot{e}}_{j} (q_{i j} t)) \\ - k_{i} ⋁_{j = 1}^{n} n_{i j} (t) f_{j} ({\dot{e}}_{j} (q_{i j} t)) - η_{0} sign (z_{i} (t)) | e_{i} (t) |^{θ} - η_{2} sign (z_{i} (t)) | z_{i} (t) |^{θ}, \end{matrix}$ and ${\begin{matrix} {\bar{u}}_{i} (t) = & - {\bar{λ}}_{i} e_{i} (t) - η_{1} sign (e_{i} (t)) | e_{i} (t) |^{θ}, \\ {\tilde{u}}_{i} (t) = & - {\tilde{λ}}_{i} z_{i} (t) - η_{0} sign (z_{i} (t)) | e_{i} (t) |^{θ} - η_{2} sign (z_{i} (t)) | z_{i} (t) |^{θ} - σ_{i} sign (z_{i} (t)) \end{matrix}$

Examples

Example 1

Consider the following system of model (1) with $i, j = 1, 2$ : $\frac{d^{2} x_{i} (t)}{d t^{2}} = - a_{i} (t) \frac{d x_{i} (t)}{d t} - b_{i} (t) x_{i} (t) + F_{i} (x (t)), t \geq t_{0} = 1,$ where $x (t) = {(x_{1} (t), x_{2} (t))}^{T}$ , $f_{j} (x_{j}) = \tanh (x_{j})$ , $c_{11} (t) = d_{11} (t) = f_{11} (t) = 0.5 \sin (t) + 0.5$ , $c_{12} (t) = d_{12} (t) = f_{12} (t) = 0.5 \cos (t) + 0.5$ , $c_{21} (t) = d_{21} (t) = f_{21} (t) = 0.25 \sin (t) + 0.25$ , $c_{22} (t) = d_{22} (t) = f_{22} (t) = 0.25 \cos (t) + 0.25$ , $h_{11} (t) = k_{11} (t) = m_{11} (t) = n_{11} (t) = 0.5 \sin (2 t) + 0.5$ , $h_{12} (t) = k_{12} (t) = m_{12} (t) = n_{12} (t) = 0.5 \cos (2 t) + 0.5$ , $h_{21} (t) = k_{21} (t) = m_{21} (t) = n_{21} (t) = 0.125 \sin (t) + 0.125$ , $h_{22} (t) = k_{22} (t) = m_{22} (t) = n_{22} (t) = 0.125 \cos (t) + 0.125$ , $g_{i j} (t) =$

Conclusions

Based on finite-time stability theory and combining with inequality techniques and some analysis methods, we have investigated the FTS for FNTINNs with time-varying coefficients and proportional delays. By designing two different types of controllers, some novel and useful FTS criteria have been established. Finally, we provided two examples and their numerical simulations to demonstrate the validity of the obtained results. It is believed that those proposed controllers and the derived results

Acknowledgement

The authors are grateful for the support of the National Natural Science Foundation of China (11601268).

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Finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays

Abstract

Introduction

Section snippets

Preliminaries

Main results

Examples

Conclusions

Acknowledgement

Physica D

Neurocomputing

Neurocomputing

Neurocomputing

Neural Netw.

Neurocomputing

Neurocomputing

Chaos Solitons Fractals

Neurocomputing

Neurocomputing

Neural Netw.

Neural Netw.

ISA Trans.

Neurocomputing

Neurocomputing

Neurocomputing

Neural Netw.

Neural Netw.

Neural Netw.

Neurocomputing

Neural Netw.

Neurocomputing

J. Assoc. Arab Univ. Basic Appl. Sci.

Neurocomputing

Nonlinear Dyn.

Nonlinear Anal. Hybrid Syst.