1 Introduction
Fractional differential calculus (FDC) has more properties than ordinary differential calculus (ODC). The main attractive application of FDC is its weak singularity, degrees of freedom, non-locality, and infinite memory [
1]. Taking arbitrary order in integrals and derivatives is an attractive usage of FDC and can be applied to various medical applications. There are many models of FDCs which are widely used in the fields of engineering and sciences. FDC plays an important role in biological sciences as well. It is used in analysis of breast cancer [
2], tumor-immune system [
3], diabetes [
4], Nipah virus [
5], and coronavirus [
6], HIV model [
7] and many more. Apart from this, applications of FDC can be seen in real-life science and engineering problems [
8], fractional fractal system [
9], the fractional ecological system [
10], heat transfer, mechanics, wide range fields, electric circuits [
11] and fractional partial differential equations [
12,
13].
In the 17th century, Christiaan Huygens described the phenomenon of synchronization. Finding an oscillating object’s rhythm from weak interaction is known as synchronization. It has been applied in pattern recognition, image processing, and object detection [
14], references therein. There are many kinds of synchronization like mutual and high-order synchronization, Kuramoto self-synchronization, complete synchronization, weak and robust synchronization, global synchronization, quasi synchronization, sampled data synchronization, global non-fragile synchronization, bipartite synchronization and references therein [
15‐
19]. Bipartite consensus means all the processing elements converge to a value and have the same modulus except for its sign [
20,
21]. A network of structurally balanced signed graphs is known as a bipartite consensus [
20]. If it is not structurally balanced, the network reaches an interval bipartite consensus [
22]. This consensus of synchronization is said to be bipartite synchronization. Over the years, several results have been established in bipartite synchronization, bipartite synchronization of Lur’e network [
23], fractional order bipartite fixed time synchronization [
24], coupled delayed neural network (CDNN) of bipartite synchronization [
19] and bipartite synchronization of memristor-based fractional-order CDNN [
25].
Leon O. Chua postulated a new circuit element. The two-terminal device, obtained by applying the initial conditions of the pinched hysteresis loop in the voltage-current (v-i) plane powered by the bipolar periodic voltage, is called a memristor. It was an experimental observation in HP (Hewlett−Packard) laps. The main advantage of a memristor is non-volatile memory. It contains many applications, such as high density, lower power, ease of integration, nano-dimension and good scalability [
26,
27]. The neural network is parallel distributed information to a processing structure. There are several types of neural networks, such as the stochastic neural network [
28], Chebyshev neural network [
10], BAM neural network [
29], discontinuous neural network [
18] and multiple memristor neural network [
30]. Delay is unavoidable in neural networks, synchronization of memristor delayed neural network, synchronization of coupled delayed neural network (CDNN), synchronization of recurrent neural networks with time varying delay [
31], memristor based inertial neural network with mixed time varying delay by using new non-reduced order method [
27], switching law with time-varying delay [
32], further refer [
15,
19,
33].
Now, FDC-based neural networks get more attention from many authors; for example, fractional neural network (FNN) [
24,
34‐
36], fractional memristor neural network (FMNN) [
37], fractional memristor delayed neural network (FMDNN) [
15], memristor-based fractional-order coupled delayed neural network (FCDNN) [
25] are widely studied. Different kinds of controllers are used in ODC and FDC. Some of the widely used ones are pinning control [
19,
23,
25], event triggered, periodic and self triggered controllers [
35,
36,
38], event triggered pinning control [
21], adaptive event triggered control [
39], impulsive control [
40‐
42], sampled data control [
17,
23], terminal and quantized slide mode control [
10,
31,
43], delay-dependent feedback controller [
27] and distributed control [
44]. The main reason for using pinning control is to reduce large network costs. It should be observed that continuous communication at every time may exist redundant and high energy consumption in control systems. It may be stopped when the event-triggered mechanism maintains the designed controller between two trigger instants. Mainly using event-triggered control reduced superfluous utilization [
21]. periodic event-triggered of singular system with cyber-attacks refer [
38], event triggered fractional-order multi-agent system [
39], event triggered of multiple fractional order neural networks with time varying delays [
35], event triggered bipartite synchronization of multi-order fractional neural networks [
36]. The Laplace transform [
2,
3,
6], the Lyapunov function [
10,
16,
18,
37], and linear matrix inequality (LMI) [
25,
34] are used in the study of FDC. Lyapunov function is classified as Lyapunov Razumikhin [
40,
41] and Lyapunov krasovskii function (LKF) [
21,
25,
28,
29,
45]. Moreover, a neuron’s activation function applied in the Lyapunov function is the Lur’e-Postnikov Lyapunov function (LPLF) [
18,
34]. It helps to reduce conservatism.
The essential novelties of this paper are the fractional order multiple memristor coupled delayed neural network (FMMCDNN) and the distributed event-triggered pinning control and descriptor method proposed by developing fractional order Jensen’s inequality to achieve the bipartite leader and leaderless synchronization. Secondly, the bipartite leader and leaderless synchronization are proven for an FMMCDNN using LPLF with LKF.
Based on the above discussions, this paper’s major contributions are as follows:
(i)
The rarely sought descriptor method in FMMCDNN with LKF of delay-dependent stability criteria is discussed here.
(ii)
This paper focuses on the bipartite synchronization of signed networks and FMMCDNN using an event-triggered pinning control with delay conditions,
(iii)
we prove that the Zeno behavior is avoided in an event triggered pinning control,
(iv)
we also prove the above three results in the case of an FMMCDNN with LPLF added to LKF.
We have formulated an FMMCDNN in Sect.
2, along with some necessary definitions. In Sect.
3, the formulation and leader bipartite synchronization of an event-triggered pinning control is discussed, and the leaderless bipartite synchronization can be seen in Sect.
4. Section
5 deals with the Lur’e Postnikov Lyapunov function criteria. And we can see numerical examples and the conclusion in Sects.
6 and
7, respectively.
Notations
sign(\(\cdot \)) represent the signum or sign function, \(\otimes \) means the Kronecker product, \(w_{p}\) denotes the gauge transformation, \(l_{pj}\) represents a Laplacian matrix, \(L^{s}\) and \(L^{u}\) are Laplacian of signed and unsigned matrices respectively, and \(\bar{co}\) means convexity closure. \(\varpi =\begin{pmatrix} P&{}Q\\ *&{}S \end{pmatrix}\), where \(*\) represent transpose of Q and \({\mathcal {G}}^{SGN}\) denotes signed network.
Consider FMMCDNN of signed network
\({\mathcal {G}}^{SGN}\) expressed by:
$$\begin{aligned} {}^{C}_{t_{0}}D^{q}_{t}\big (x_{i} (t) \big )= & {} -A x_i(t)+ B\big (x_{i}(t) \big ) f \big (x_{i}(t) \big )+ C\big (x_{i}(t) \big ) g \big (x_{i} \big (t-\tau (t) \big ) \big )\nonumber \\{} & {} -\sigma \sum _{j=1}^{N} \big |a^{s}_{ij} \big | \big (x_{i} \big (t_{k} \big )-sign \big (a^{s}_{ij} \big ) x_j \big (t_{k}) \big )+u_{i}, \end{aligned}$$
(5)
where the state variable
\(x_{i}(t),(t\ge 0)\),
\(x_{i}(t)=(x_{i1}(t), \ldots x_{in}(t))^{T}\) (capacitor’s voltage) is the
ith neuron,
\(i=\{1,2, \ldots N\}\), diagonal matrices of neurons denoted by
\(A=diag(a_{1},a_{2}, \ldots a_{n})\). The connective weighted memristor matrices of
B &
C are defined as follows,
\(B(x_{i}(t))=[b_{gj}(x_{ij}(t))]_{N\times N}\) and
\(C(x_{i}(t))=[c_{gj}(x_{ij}(t))]_{N\times N}\) respectively. Let
f &
g are bounded feedback functions with and without delay, and
\(f(x_{i}(t))=(f_{1}(x_{i1}(t)), \ldots f_{n}(x_{in}(t)))^{T}\) and
\(g(x_{i}(t-\tau {(t)}))=(g_{1}(x_{i1}(t-\tau {(t)})), \ldots g_{n}(x_{in}(t-\tau {(t)})))^{T}\). The coupling strength of the network is represented by
\(\sigma \), and
\(\tau (t)\) stands for bounded and non-differentiable node delay
\(0\le \tau (t)<\bar{\tau }\). Using a memristor instead of a resister in a network forms a memristor neural network. The memristor neural network with a coupled term forms a multiple memristor neural network [
26,
30]. If a coupling is symmetric, the adjacency matrix satisfies
\(a_{ij}=a_{ji}\). But it is not necessarily symmetric. If signed network
\({\mathcal {G}}^{SGN}\) (
5) is strongly connected and
\(a_{ij}\) is positive then the coupling term
\(a^{s}_{ij}(x_{i}(t)-x_{j}(t))\), otherwise -
\(a^{s}_{ij}(x_{i}(t)+x_{j}(t))\). Let the initial condition of
\(x_{i}(t)\) be defined as
\(x_{i}(t)=\phi _{i}(t), t\in [-r,0]\), where
\(r=\sup \limits _{t\ge 0}\{\tau (t)\}\). The initial condition
\(\phi _{i}(t)\), which belongs to the bounded feedback functions on
\([-r,0]\). The leader node of network (
5) is defined as
$$\begin{aligned} {}^{C}_{t_{0}}D^{q}_{t}(s_{i}(t))=-A s_i(t)+ B(s_{i}(t)) f(s_{i}(t))+ C (s_i(t)) g(s_{i}(t-\tau (t))), \end{aligned}$$
(6)
\(b_{gj}(s_{ij}(t))\) and
\(c_{gj}(s_{ij}(t))\) represent the memristor connective weights of matrices, where
\(s_{i}(t)=(s_{i1}(t), \ldots s_{in}(t))^{T}\).
3.1 Scheme of Event Triggered Pinning Control Formulation
The impulsive instant
\(t_{k}\) is defined by
$$\begin{aligned} t_{k+1} = \inf \left\{ t>t_{k} \mid \ g_{i}(t)\ge 0 \right\} ,\quad k\in {\mathbb {N}}. \end{aligned}$$
If
\(\Gamma =\{\Gamma _{1},\Gamma _{2},\Gamma _{3}\}\) be positive definite matrix and
\(0\le \zeta _{1},\zeta _{2}<1\), define
\(\tau (t)=t-t_{k}\in [t_{k},t_{k+1})\), then triggering function is defined as follows,
$$\begin{aligned} g_{i}(t)= & {} \varpi ^{T}_{i}(t)\Gamma _{1}\varpi _{i}(t) - \zeta _{1} \big [\varpi ^{T}_{i}(t)+e_{i}^{T}(t-\tau (t)) \big ]\Gamma _{2} \big [\varpi _{i}(t)+e_{i}(t-\tau (t)) \big ]\nonumber \\{} & {} -\zeta _{2}e_{i}^{T}(t-\tau (t))\Gamma _{3}e_{i}(t-\tau (t))\ge 0 \end{aligned}$$
(10)
where triggering error
\(\varpi _{i}(t)=e_{i}(t_{k})-e_{i}(t)\).
Now, we define the event triggered pinning controller \(u_{i}(t)=-\sigma d_{i}(x_{i}(t_{k})-w_{i}s_{i}(t_{k}))\), \(t\in [t_{k},t_{k+1})\), where \(d_{i}\) is the pinning feedback gain. If \(d_{i}\) is positive then the vertices are pinned, otherwise the value is zero. If \(w_{i}=1\) then \(i\in V_{1}\), and if \(w_{i}=-1\) then \(i\in V_{2}\). Suppose, if \(w_{i}=I_{N}\), bipartite synchronization changes into the traditional leader-follower synchronization.
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