Asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints

doi:10.1016/j.cnsns.2022.106694

Communications in Nonlinear Science and Numerical Simulation

Volume 114, November 2022, 106694

https://doi.org/10.1016/j.cnsns.2022.106694 Get rights and content

Highlights

•
This paper considers the influence of algebraic constraints on FDMNNs.
•
A new stability criterion of FSDMNNs is attained by a novel Lyapunov functional.
•
Two easy-to-verify synchronization conditions are gained by suitable controllers.
•
Our results can be deemed an extension of the existing ones.

Abstract

The asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints in Riemann–Liouville sense will be investigated in this article. First, algebraic constraints are introduced for the first time into the existing fractional delayed memristive neural networks, and a new fractional singular delayed memristive neural networks (FSDMNNs) model is presented. Then, within the framework of Filippov’s solution, a less conservative result for the asymptotic stability of FSDMNNs is obtained by Lyapunov–Krasovskii functional. Subsequently, the appropriate feedback scheme and adaptive scheme are designed to synchronize FSDMNNs and two sufficient conditions are acquired. In addition, the results not only address the influence of delays and algebraic constraints, but can also easily detect and synchronize the actual memristive neural networks. Finally, numerical simulations frankly confirm the correctness and validity of the derived results.

Introduction

Chua first predicted in 1971 that in addition to resistance, capacitance and inductance, there would be a fourth type of basic passive electronic device, called memristor [1]. However, HP Labs prepared a physical memristor element based on $T_{i} O_{2}$ for the first time in 2008 [2] and caused a sensation in the field of international electronic technology. As a new type of information storage and processing device, it has been proven to be an ideal component for simulating neural synapses in human brain neural networks due to its nano-scale size, fast switching and low power consumption [3], [4]. Furthermore, the unique memory properties of memristor have deeply penetrated into the fields of on-chip learning [5], [6], neural networks (NNs) [7] and optimization problems [8].

Recently, the dynamical properties of memristor-based neural networks (MNNs) have become an intensive research hotspot, and related achievements such as chaos [9], passivity [10] and synchronization [11], [12] have emerged one after another. Stability, as the most basic dynamic indicator for the long-term stable operation of the system, has attracted more and more attention [13], [14], [15]. And yet, in the hardware implementation process of MNNs, due to the limited switching speed of electronic devices and the transmission speed of electrical signals, the system will inevitably have delay effects and may disrupt the stability behavior of MNNs [16]. Therefore, the stability research of delayed memristive neural network (DMNNs) is profitable, which leads to many remarkable results [17], [18], [19].

Essentially, synchronization can be deemed as the stability analysis of the error system. In addition, as the most symbolic collective behavior in practical applications, it is active in communication [20], image encryption [21], NNs [22], [23], etc. For the synchronization problems of DMNNs, many excellent results have also been reported, as regards adaptive synchronization [24], exponential synchronization [25] and global synchronization [26]. In practice, the implantation of controllers can readily and effectively achieve the synchronization of DMNNs. And many related control strategies, such as event-triggered control [27], hybrid impulsive control [28], sliding mode control [29], have been designed.

The above-mentioned DMNNs works utilize integer-order differential systems to mathematically model their dynamic performance. Nonetheless, the integer-order differential systems can not describe the memory characteristics of neurons and their dependence on past history, while the fractional differential operator embodies all the information from the beginning to the current time, which can more precisely describe the “infinite-memory” of neurons. In fact, fractional calculus has been successfully applied to DMNNs model, namely, fractional delayed memristive neural networks (FDMNNs). Their dynamic behaviors were discussed in [30], [31], [32], [33]. Ali et al. [30] mainly investigated the finite-time stability of FDMNNs. Wang et al. [31] paid special attention to its synchronization behavior. Moreover, Bao et al. [32] showed great interest in non-fragile state estimation.

Nevertheless, the current research results on DMNNs or FDMNNs rarely take constraints into account. It should be noted that constraints generally exist in practical applications, as for security protection [34], saturation [35]. Thus, the dynamic analysis of the NNs model is fully requisite to think about the constraints. Relevant achievements have sprung up over recent years [36], [37], [38], [39]. Undeniably, certain neuron chains need to satisfy some algebraic constraints in NNs before they can be activated. And algebraic constraints have been drawn into the discussion of NNs [40], [41]. The stability of discrete-time NNs under algebraic constraints was discussed in [42] by using run-to-run controllers. With the help of input disturbance estimation and stochastic technique, [43] studied the robust stabilization of NNs with algebraic constraints and Markov jump. Moreover, under algebraic constraints, Zhang et al. [44] researched the adaptive sliding mode control problem of nonlinear fractional systems with mismatched uncertainties. From this fact, it is substantial to consider the influence of algebraic constraints in the qualitative analysis of FDMNNs. But, little work has been done in this direction so far. In fact, taking into account fractional calculus, algebraic constraints, and time delays, the considered mathematical model is regarded as a class of fractional singular delayed memristive neural networks (FSDMNNs). It frequently has some special properties, such as regular and impulsive, which do not need to be considered in application systems. Under regular and impulse-free, a series of conclusions of DMNNs and FDMNNs can not be simply converted into FSDMNNs. Therefore, it is hard to explore the dynamic characteristics of such complex systems. More specifically, the stability and synchronization analysis of FDMNNs with algebraic constraints are still unsolved and challenging.

Derived from the above discussion, the goals of this paper are to attempt to analyze the stability and synchronization of FSDMNNs with the following contributions:

(1) Different from the previous research on [18], [30], [33], this paper will consider the influence of algebraic constraints on FDMNNs, and a new class of FSDMNNs model is proposed.

(2) A new algebraic criterion for the asymptotic stability of FSDMNNs is acquired by Lyapunov direct method.

(3) Two suitable control schemes are adopted to synchronize FSDMNNs, and two easy-to-verify algebraic inequality conditions for asymptotic synchronization are gained.

(4) Our results can be deemed an extension of the existing ones.

The rest of this article is arranged as follows. Preliminaries and model description are introduced in Section 2. Section 3 are the main results. In Section 4, simulations verify the availability of the acquired results. Section 5 is the conclusion and prospect of this paper.

Table 1 summarizes some symbols commonly used in this paper to ensure the fluency of the paper:

Section snippets

Preliminaries

Definition 2.1

[45]

The Riemann–Liouville-type fractional integral and derivative of $μ (t) \in C^{1} ((t_{0}, + \infty), R)$ are respectively defined as: $D^{- q} μ (t) = \frac{1}{Γ (q)} \int_{t_{0}}^{t} {(t - s)}^{(q - 1)} μ (s) d s, (q > 0);$ $D^{q} μ (t) = \frac{1}{Γ (n - q)} \frac{d^{n}}{d t^{n}} \int_{t_{0}}^{t} \frac{μ (s)}{{(t - s)}^{q + 1 - n}} d s, (n - 1 < q < n) .$

Property 2.1

[45]

If $p > q > 0$ , then inequality holds for $μ (t) \in C^{1} ((t_{0}, + \infty), R)$ : $D^{q} (D^{- p} μ (t)) = D^{q - p} μ (t)$ .

Lemma 2.1

[45]

Let $μ (t) : R^{n} \to R^{n}$ be a vector of differentiable function. Then: $\frac{1}{2} D^{q} (μ^{T} (t) μ (t)) \leq μ^{T} (t) D^{q} μ (t), \forall q \in (0, 1)$ .

Lemma 2.2

[46]

$\forall$ $γ_{1}, γ_{2} \in R^{n}$ , $ɛ > 0$ , inequality $2 γ_{1}^{T} γ_{2} \leq ɛ γ_{1}^{T} γ_{1} + \frac{1}{ɛ} γ_{2}^{T} γ_{2}$ holds.

Lemma 2.3

[47]

A scalar $ρ$ and a vector function $x (t) : [t - ρ, t] \to R^{n}$ such that the

Asymptotic stability of FSDMNNs

This section will focus on the asymptotic stability of FSDMNNs with the assistance of traditional Lyapunov functional. And the detailed discussion can be seen in the following theorem.

Theorem 3.1

When Assumption 2.1 holds, then the FSDMNNs (1) will be asymptotic stability if the following inequality satisfies: $- c_{i} + \frac{1}{2} \sum_{j = 1}^{n} [{\tilde{a}}_{i j} + {\tilde{a}}_{j i} δ_{1 i}^{2} + {\tilde{b}}_{i j} + {\tilde{h}}_{i j} + \frac{1}{1 - σ} ({\tilde{b}}_{j i} δ_{2 i}^{2} + τ^{2} {\tilde{h}}_{j i} δ_{3 i}^{2})] < 0 .$

Proof

Pick the following Lyapunov functional: $V (t) = V_{1} (t) + V_{2} (t) + V_{3} (t),$ where: $V_{1} (t) = \frac{1}{2} \sum_{i = 1}^{n} e_{i} D^{q - 1} (μ_{i}^{2} (t)),$ $V_{2} (t) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} \frac{{\tilde{b}}_{j i} δ_{2 i}^{2}}{2 (1 - σ)} \int_{t - τ_{1} (t}$

Numerical examples

Three illustrative examples are given to support the effectiveness of the obtained results in this part.

Example 4.1

Consider the stability of following FSDMNNs with three neurons: $e_{i} D^{q} μ_{i} (t) = - c_{i} μ_{i} (t) + \sum_{j = 1}^{3} a_{i j} (μ_{i} (t)) f_{1 j} (μ_{j} (t)) + \sum_{j = 1}^{3} b_{i j} (μ_{i} (t)) f_{2 j} (μ_{j} (t - τ_{1} (t))) + \sum_{j = 1}^{3} h_{i j} (μ_{i} (t)) \int_{t - τ_{2} (t)}^{t} f_{3 j} (μ_{j} (s)) d s,$ where $i = 1, 2, 3$ ; $q = 0.7$ ; $e_{1} = 2$ , $e_{2} = 3$ , $e_{3} = 0$ ; $c_{1} = 7$ , $c_{2} = 8$ , $c_{3} = 9$ . Let time delays: $τ_{1} (t) = 0.4 \frac{e^{t}}{1 + e^{t}}$ , $τ_{2} (t) = 0.7 τ_{1} (t)$ , which signifies: $σ = 0.12$ , $τ = 0.4$ . The activation functions are: $f_{1 j} (μ_{j} (\cdot)) = t a n h (μ_{j} (\cdot))$ , $f_{2 j} (μ_{j} (\cdot)) = f_{3 j} (μ_{j} (s)) = s i$

Conclusion

In this paper, we proposed a new class of FDMNNs with algebraic constraints (i.e. FSDMNNs) in the sense of Riemann–Liouville, and investigated its asymptotic stability and synchronization. First, under the help of the Lyapunov direct method and inequality technique, we obtained a concise algebraic criterion for the asymptotic stability of FSDMNNs. Thereafter, two control schemes are submitted, two sufficient and easy-to-verify algebraic inequality results are obtained to ensure the

CRediT authorship contribution statement

Xiang Wu: Conceptualization, Methodology, Writing – original draft and subsequent revisions. Shutang Liu: Fund support, Writing guidance. Huiyu Wang: Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Nos. U1806203 and 61533011).

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Research paperAsymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints

Highlights

Abstract

Introduction

Section snippets

Preliminaries

[45]

[45]

[45]

[46]

[47]

Asymptotic stability of FSDMNNs

Numerical examples

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Appl Math Comput

Neurocomputing

Neural Netw

Neurocomputing

Appl Math Comput

Neural Netw

Appl Math Comput

Appl Math Comput

ISA Trans

Neural Netw

Neural Netw

Neural Netw

Comput Biol Med

Appl Math Comput

Appl Math Lett

Neurocomputing

Memristor-the missing circuit element

IEEE Trans Circuit Theory

The missing memristor found

Nature

Cortical computing with memristive nanodevices

SciDAC Rev

Nanoscale memristor device as synapse in neuromorphic systems

Nano Lett

On-chip error-triggered learning of multi-layer memristive spiking neural networks

IEEE J Emerg Sel Top Circuits Syst

Memristive quantized neural networks: A novel approach to accelerate deep learning on-chip

IEEE Trans Cybern

Neuromorphic hardware system for visual pattern recognition with memristor array and CMOS neuron

IEEE Trans Ind Electron

Exponential H∞ state estimation for memristive neural networks: Vector optimization approach

IEEE Trans Neural Netw Learn Syst

Compound synchronization based on memristive cellular neural network of chaos system

J Comput Nonlinear Dyn

Exponential passivity of memristive neural networks with time delays

Neural Netw : Official J Int Neural Netw Soc

Research paper
Asymptotic stability and synchronization of fractional delayed memristive neural networks with algebraic constraints

Exponential $H_{\infty}$ state estimation for memristive neural networks: Vector optimization approach