Abstract
The finite-time Mittag-Leffler stability for fractional-order quaternion-valued memristive neural networks (FQMNNs) with impulsive effect is studied here. A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed. In quaternion field, by using the framework of Filippov solutions as well as differential inclusion theoretical analysis, suitable Lyapunov-functional and some fractional inequality techniques, the existence of unique equilibrium point and Mittag-Leffler stability in finite time analysis for considered impulsive FQMNNs have been established with the order \(0<\beta <1\). Then, for the fractional order \(\beta \) satisfying \(1<\beta <2\) and by ignoring the impulsive effects, a new sufficient criterion are given to ensure the finite time stability of considered new FQMNNs system by the employment of Laplace transform, Mittag-Leffler function and generalized Gronwall inequality. Furthermore, the asymptotic stability of such system with order \(1<\beta <2\) have been investigated. Ultimately, the accuracy and validity of obtained finite time stability criteria are supported by two numerical examples.
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1 Introduction
In 1695, the foundation of non-integer order calculus, which is a generalization integer order differential and integrals was first of all discussed through Guillaume de Leibnitz and Gottfried Wilhelm Leibnitz, and its development were inch by inch for long period. Until recently, it has been a great research topic due to the fact many fractional order models play a crucial role in many real world objects. Comparing to an integer order dynamical model, fractional order dynamical model is more accuracy, non-local and has weakly singular kernels but integer order dynamical behavior fails in this aspect. From the application perspective, an electronic implementation of an artificial neural network model, many researchers have combined the fractional order calculus into neural networks to look at the fractional order neural network model (FONNs). Currently, fractional order calculus has been very promising areas of research and thus successfully applied in both theoretical and applicable manners [1,2,3,4]. It is well known that stability is the primary condition of the several systems [5,6,7,8]. At present, the stability analysis of neural networks and differential equations models become a hot topic and some excellent has been reported, see [9,10,11,12,13,14,15,16]. In [14], the authors investigated the stability criteria of Riemann–Liouville sense fractional order impulsive fuzzy neural networks with delay, and proposed the global asymptotic stability analysis by using fractional Barbalat’s lemma and Lyapunov stability theory. In [17], the author researched stability analysis of fractional order delayed neural networks and several conditions to ensure the existence, uniqueness and finite time stability were established based Gronwall’s inequality, method of iteration and contraction mapping principle. In [18], the authors demonstrated the finite time stability analysis of fractional order neural networks by means of estimates of Mittag-Leffler functions, generalized Gronwall’s inequality and Laplace transform.
The idea of memristor has been analyzed thinking about that 1971, at the same time as Leon Chua has proposed for the first time in a properly-organized and mathematically described manner [19]. Despite the fact that, the concept of memristor-like gadgets has been counseled in advance in 1960 by way of Bernard [20], Leon Chua changed into the primary one not simplest to offer a possible foundation for memristor existence, however also to estimate and mathematically describe its meant conduct and residences. Almost after 40 years for memristor to change from a definitely theoretic idea into workable usage. In 2008 a gathering of researchers from Hewlett-Packard Labs lead through Stan Williams has finally formulated by means of practically working memristor [21]. In [22], Kim et al., was successfully initiated by Memristor bridge synapse architecture and resolve the difficulty of the problem of nonvolatile synaptic weight garage and put in force a recently proposed hardware learning techniques. After this memristor has found various applications in numerous interdisciplinary field [23,24,25]. In general, fractional order memristor based neural networks model (FMBNNs) is an improved fractional order neural networks model by traditional resistor replaced by memristors. Many authors have investigated the several dynamical behaviors of fractional-order memristor based neural networks. In [26], the authors applied a Holder inequality to analyze the finite stability of fractional order delayed memristive complex-valued neural networks with order, both \(0<\beta <0.5\) and \(0.5\le \beta <1\), respectively. In [27], Rakkiyappan et al. has deliberated the finite time stability of fractional order complex valued neural networks with fractional order \(1<\beta <2\) by using generalized Gronwall inequality and Holder inequality.
Quaternion algebra is a standout amongst the most renowned type and its universal extension case of real-valued and complex-valued numbers, which became first of all originated by means of Hamilton in 1843 [28]. Recently, Quaternion-valued neural networks (QVNNs) has attracted considerable attention owing to its widespread applications in various disciplines in attitude control, satellite tracking, image processing, computer graphics, three dimensional modelling, four dimensional modelling and extensively investigated by many researchers, see [29,30,31,32,33,34,35] for instance. Comparing to real-valued MNNs and complex-valued MNNs, Quaternion-valued memristive neural networks (QVMNNs) is more storage capacity and complicated properties and it consists quaternion memristive connection weights, system state, and neuron activation functions. In color image compression, real-valued MNNs and complex-valued MNNs may be likewise used to transmit the colour signals yet with moderately poor impact. Truth be told, by means of three primary colours red, green and blue can be changed over into three signals about the three essential colors with certain proportion which may be transmitted by means of three channels i, j and k of the QVMNNs, and afterward changed over into the colour images too. However, real-valued MNNs and complex-valued MNNs can’t understand this ideal impact. As a result of the non-commutativity, conventional techniques used to investigate the stability of real-valued MNNs and complex-valued MNNs cannot be directly applied to the similar problems of QVMNNs. In this manner, the investigation on the fractional order QVMNNs dynamical behaviors in both theory and applications has turned out to be urgent and mandatory. Consequently, the dynamics of integer order QVNNs have been taken into consideration by means of many research scholars and a large number of great outcomes has been gained in the existing literature [36,37,38,39]. But there is little attention about the dynamics of FQNNs have been found in the existing literatures. For example, the authors in [40] presented the global Mittag-Leffler stability and global Mittag-Leffer synchronization analysis of FQNNs with linear threshold neurons by using matrix eigenvalue, M-matrix theory and Lyapunov method. In [41], the robust asymptotical stability and robust asymptotical synchronization of memristor based fractional order QVMNNs with time delays and parameter uncertainties by using nonsmooth analysis, fractional order comparison principle and Lyapunov direct method.
On the other hand, many physical processes are distinguished by abrupt changes at certain moments of time in the real-world problems. These abrupt changes were mentioned as impulsive phenomena. These impulses can influence the dynamical performance of the system trajectory from original direction in a moment [42,43,44]. Hence, the dynamical behaviours of FONNs might be described more accurately by considering the impulse. Moreover, the impulsive fractional-order neural networks showed more advanced in describing the hereditary and memory properties for various materials and processes comparing to the impulsive integer order neural networks. Along these lines, the investigation on the dynamic behavior of FONNs with impulsive effects becomes more essential ones and some excellent results have been devoted in more as of late [45,46,47]. For example in [45], the authors gave some existence, uniqueness and Mittag-Leffler stability criteria for impulsive FONNs in terms of linear matrix inequality based on topological degree properties and positive definite quadratic Lyapunov function. In [46], by using contraction mapping principle, the linear growth condition of activation function and positive definite quadratic Lyapunov function, the author have investigated about the global Mittag-Leffler stability of FONNs with one side Lipschitz condition and impulsive effects. In [47], by means of contraction mapping principle, fractional order comparison principle and fractional order absolute valued Lyapunov functional with one norm, the global asymptotical stability of a class of impulsive FONNs in complex field was demonstrated. Limin et al. [47] investigated the asymptotic stability of impulsive delayed fractional order complex valued neural networks with order \(0<\beta <1\) by fractional order comparison principle and Lyapunov functional. However, there are few articles focused on the stability analysis of memristor based neural networks with impulsive effects. To the best of author’s knowledge, nevertheless, Mittag-Leffler finite time stability analysis of fractional order impulsive QVMNNs dynamical behaviours has not been investigated yet.
Sparked by the above reason and discussion, we try to investigate the finite-time Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with order \(0<\beta <1\) and \(1<\beta <2\), the problem remains open, and is no article in existing literature. Consequently, we can try to remedy this hard and essential problem. The main challenge and contribution of this work are highlighted in the following aspects:
- 1.
A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed.
- 2.
The new brand of novel sufficient criterion proposed first to ensure the existence and finite time Mittag-Leffler stability of impulsive FQMNNs with order \(0<\beta <1\) by means of Banach contraction mapping principle and fractional order Lyapunov functional.
- 3.
When \(\beta \) satisfying \(1<\beta <2\) and the model at the absence of impulsive effects, the finite time stability criteria are introduced by using Laplace transform, Mittag-Leffler function and generalized Gronwall inequality.
- 4.
As some special cases of proposed results, we also investigate the asymptotic stability of FQMNNs with fractional order \(1<\beta <2\).
- 5.
Most of the FNNs have not now taken into consideration quaternion memristive connection weights, system state, and neuron activation, especially FNNs model, however, our results make it up.
The rest of the proposed work is furnished as follows: In Sect. 2, the basic concepts of quaternion algebra, some necessary definitions about fractional order calculus are listed. Further, some necessary assumptions and finite-time Mittag-Leffler stability definitions together with a few beneficial lemmas needed in this paper are given. The main results with order \(0<\beta <1\) and \(1<\beta <2\) are established in Sect. 3. Two numerical examples and their computer simulations are provided to illustrate the effectiveness of the acquired results in Sect. 4. At last, Sect. 5 ends with conclusions.
2 Preliminaries
In this section, we will recall some basic knowledge of quaternion algebraic concepts and fractional order calculus. In addition, some lemmas and problem statement are presented, which serve for the following sections.
2.1 Quaternion Algebra
As a type of super complex number, quaternion consists a real part and three imaginary parts, and a real quaternion or quaternion y can be expressed as:
where \(h,q,w,z\in \mathbb {R}\), and the imaginary roots i, j, k satisfy the Hamilton multiplication rules:
From the above Hamilton rules, the quaternion multiplication is non commutative. The quaternion set is denoted by:
\(\mathbb {Q}^{m}\) signify the set of all m dimensional quaternion space. The operation of addition and subtraction in quaternion field are similar as those in complex numbers or vectors, by
where \(y=h+iq+jw+kz\) and \(z=\tilde{h}+i\tilde{q} +j\tilde{w}+k\tilde{z}\). According to Hamilton multiplication rules (1), the product of yz is described as:
The absolute values of y is described by:
For a quaternion valued function x(t) is denoted by \(y(t)=h(t)+iq(t)+jw(t)+kz(t)\), where \(h(t),\;q(t),\;w(t),\;z(t)\) are all real-valued function. Furthermore, the norm of y of vector quaternion \(y=\big (y_1,\ldots ,y_m\big )^T\in \mathbb {Q}^m\) is given by
2.2 Basic Tools of Fractional Calculus
Definition 2.1
[48] The Riemann–Liouville fractional integral of y(t) is defined as:
where \(\beta \in \mathbb {R}^{+}\).
Definition 2.2
[48] The Caputo-type fractional integral of y(t) is defined as:
where \(\beta \in \mathbb {R}^{+},\;m\in \mathbb {Z}^{+}\).
Proposition 1
[48] The linearity of Caputo-type fractional derivation is defined by
Definition 2.3
[48] The Mittag-Leffler function with two parameter is defined as
where \(\beta ,\;\sigma \in \mathbb {R}^{+},\;z\in \mathbb {C}\).
Definition 2.4
[48] For \(m-1<\beta <m\), the Laplace transform of the Mittag-Leffler function with two parameter:
where s and t are both variables in Laplace domain and time domain, respectively.
Lemma 2.5
[48] When \(\beta \in (0,2),\;\sigma >0\) and \(\omega \in (\frac{\beta \pi }{2},\min \{\beta \pi ,\pi \})\), then there exist two known positive scalars \(\Lambda _1>0,\;\Lambda _2>0\), such that
where \(|arg(z)|\le \omega \), \(|z|\ge 0\).
Lemma 2.6
[49] Let g(t) and u(t) are locally integrable and non negative function on the interval [0, b) and \(d(t)\le F\) defined on [0, b), where \(F>0\) is a constant. If \(\beta >0\) and the following relationships hold:
then we have
if u(t) is non decreasing on [0, b), then g(t) satisfies
Lemma 2.7
[50] If y(t) is the continuously derivable function, the following relationship true almost everywhere:
Lemma 2.8
[51] When \(\beta \in [0,2),\;\sigma >0\) and matrix A is diagonal stable, then there exist greatest eigenvalues of A, namely \(\varsigma \), such that
If \(\sigma =1,2,\beta \) and \(0\le \beta \), then
Lemma 2.9
[52] Let H(t) be a continuous derivable function on \([0,+\infty )\) satisfying
for constants \(\varepsilon _{1},\;\varepsilon _{2}\ge 0\), then
2.3 Problem Statement
In this paper, we consider a class of fractional-order quaternion-valued memristive neural networks (FQMNNs) with impulsive effects described by:
where \(p,s\in \{1,2,\ldots ,m\},\;t\ge 0\), \(D^{\beta }\) is the Caputo fractional derivative of order \(\beta \;(0<\beta <1)\), \(y_{p}(t)\in \mathbb {Q}\) signifies the state vector of the pth neuron at time t, \(a_{i}\) signifies the self feedback connection weights of pth neurons, \(L_{p}(t)\in \mathbb {Q}\) is time-varying external inputs, \(f_{s}\big (y_{s}(t)\big )\) stands for nonlinear quaternion-valued activation function of the sth neurons at time t, \(u_{ps}\big ( y_{s}(t)\big )\) is quaternion-valued memristive connection strengths, that can be discontinuous. The impulsive moment \(t_\tau ,\;\tau =1,2,\ldots \) satisfy \(0<t_1<t_2,\ldots \), \(\lim _{t\rightarrow \infty }t_\tau =+\infty \), \(y_{p}(t^{+}_{\tau })=\lim _{t\rightarrow t^{+}_{\tau }}y_{p}(t)\) and \(y_{p}(t^{-}_{\tau }) = \lim _{t\rightarrow t^{-}_{\tau }}y_{p}(t)\) are the right and left limits of \(y_{p}(t_{\tau })\), respectively. Without loss of generality we assume that, \(y_{p}(t_{\tau })=y_{p}(t^{-}_{\tau })\), which implies the solution of FQMNNs (2) is left continuous at time \(t_\tau \), the initial states of FQMNNs (2) is describes as \(y_{p}(0)=y_{p0}\).
Let
and \(\partial \Upsilon \) stands for the boundary of domain (3). The memristive connection weight is defined by:
for \(p,s\in \{1,2,\ldots ,m\}\), where \(F^{R}_{s},\;F^{I}_{s}, \;F^{J}_{s},\;F^{K}_{s}\) are known positive constants, \(\hat{u}_{ps},\check{u}_{ps}\in \mathbb {Q}\).
Since, the memristive connection strength of FQMNNs (2) is the sense of discontinuity form. As a result, the traditional solution for fractional order differential equations does not suitable to FQMNNs (2). In this case, we need to study the concept of Filippov solutions of considering the fractional order discontinuous right-hand side system.
By means of set valued mapping analysis and differential inclusion theory [53], FQMNNs (2) can be written as follows:
Let \(y_{s}=h_s+iq_s+jw_s+kz_s\in \mathbb {Q}\) and \(f_{s}(y_s)\) can be expressed by splitting into its a real part and three imaginary parts as follows:
Moreover, let \(u_{ps}(y_s)=u^{R}_{ps}(y_s)+iu^{I}_{ps}(y_s) +ju^{J}_{ps}(y_s) +ku^{K}_{ps}(y_s)\), \(\hat{u}_{ps}(y_s) =\hat{u}^{R}_{ps}(y_s)+i\hat{u}^{I}_{ps}(y_s)+j\hat{u}^{J}_{ps}(y_s) +k\hat{u}^{K}_{ps}(y_s)\), \(\check{u}_{ps}(y_s)=\check{u}^{R}_{ps} (y_s) +i\check{u}^{I}_{ps}(y_s)+j\check{u}^{J}_{ps}(y_s) +k\check{u}^{K}_{ps}(y_s)\). Then, we have
for \(p,s=1,2\ldots ,n\). Let \(y_{p}=h_p+iq_p+jw_p+kz_p\in \mathbb {Q}\), FQMNNs (2) can be expressed as follows:
By using differential inclusion of (8)–(11), one has
Equivalently, there exist \(\lambda ^{R}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{R}_{ps}\}\big ( \cdot \big )\), \(\lambda ^{I}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{I}_{ps}\}\big ( \cdot \big )\), \(\lambda ^{J}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{J}_{ps}\}\big ( \cdot \big )\) and \(\lambda ^{K}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{K}_{ps}\}\big ( \cdot \big )\) such that
In order to prove our stability results, for FQMNNs (2), we need the following assumptions and Lemma.
Assumption
\([\mathcal {A}_{1}]\) For any \(y_{s}\in \partial \Upsilon \), \(f_s(y_s)=0\).
Assumption
\([\mathcal {A}_{2}]\) For any \(y_1=h_1+iq_1+jw_1+kz_1,\;y_2 =h_2+iq_2+jw_2+kz_2\in \mathbb {Q}\) and \(s\in \{1,2,\ldots ,m\}\), there exist positive scalars \(\Phi ^{R1}_{s},\;\Phi ^{R2}_{s}, \;\Phi ^{R3}_{s},\;\Phi ^{R4}_{s}\), \(\Phi ^{I1}_{s},\;\Phi ^{I2}_{s},\;\Phi ^{I3}_{s},\;\Phi ^{I4}_{s}\), \(\Phi ^{J1}_{s},\;\Phi ^{J2}_{s},\;\Phi ^{J3}_{s},\;\Phi ^{J4}_{s}\), \(\Phi ^{K1}_{s},\;\Phi ^{K2}_{s},\;\Phi ^{K3}_{s},\;\Phi ^{K4}_{s}\) such that
Remark 2.10
Let \(f^{R}_s(y_s)=\tilde{f}^{R}_{s}\big ( h_{s},q_{s},w_{s},z_{s}\big )\), \(f^{I}_s(y_s)=\tilde{f}^{I}_{s}\big ( h_{s},q_{s},w_{s},z_{s}\big )\), \(f^{J}_s(y_s)=\tilde{f}^{J}_{s}\big ( h_{s},q_{s},w_{s},z_{s}\big )\) and \(f^{K}_s(y_s)=\tilde{f}^{K}_{s}\big ( h_{s},q_{s},w_{s},z_{s}\big )\). Assumption \([\mathcal {A}_{2}]\) holds if and only if \(\tilde{f}^{R}_{s}\big ( \pm F^{R}_{s},q_{s},w_{s},z_{s}\big ) =\tilde{f}^{I}_{s}\big (h_{s}, \pm F^{I}_{s},w_{s},z_{s}\big ) =\tilde{f}^{J}_{s}\big (h_{s}, q_{s},\pm F^{J}_{s},z_{s}\big ) =\tilde{f}^{K}_{s}\big (h_{s}, q_{s},w_{s},\pm F^{K}_{s}\big )\equiv 0\), for any \(h_{s},q_{s},w_{s},z_{s}\in \mathbb {R}\).
Remark 2.11
The first order partial derivatives of \(\tilde{f}^{R}_{s}\big (\cdot ,\cdot ,\cdot ,\cdot \big )\), \(\tilde{f}^{I}_{s}\big (\cdot ,\cdot ,\cdot ,\cdot \big )\), \(\tilde{f}^{J}_{s}\big (\cdot ,\cdot ,\cdot ,\cdot \big )\), \(\tilde{f}^{K}_{s}\big (\cdot ,\cdot ,\cdot ,\cdot \big )\) with respect to h, q, w, z exists and are continuous and bounded, that is, there exist positive constants \(\Phi ^{R1}_{s},\;\Phi ^{R2}_{s},\;\Phi ^{R3}_{s},\;\Phi ^{R4}_{s}\), \(\Phi ^{I1}_{s},\;\Phi ^{I2}_{s},\;\Phi ^{I3}_{s},\;\Phi ^{I4}_{s}\), \(\Phi ^{J1}_{s},\;\Phi ^{J2}_{s},\;\Phi ^{J3}_{s},\;\Phi ^{J4}_{s}\), \(\Phi ^{K1}_{s},\;\Phi ^{K2}_{s},\;\Phi ^{K3}_{s},\;\Phi ^{K4}_{s}\) such that
Therefore, Assumption \([\mathcal {A}_{2}]\) satisfied by means of mean value theorem for multi-variable functions.
Lemma 2.12
Under Assumptions \([\mathcal {A}_{1}]\) and \([\mathcal {A}_{2}]\), for any \(y_1=h_1+iq_1+jw_1+kz_1,\;y_2=h_2+iq_2+jw_2+kz_2\in \mathbb {Q}\), there is \(\tilde{f}^{R}_{s}\big ( \pm F^{R}_{s},q_{s},w_{s},z_{s}\big )=0\), then
where \(p,s\in \{1,2,\ldots ,m\}\), \(u^{R}_{ps}=\max \{|\hat{u}^{R}_{ps}|,|\check{u}^{R}_{ps}|\}\), \(u^{I}_{ps}=\max \{|\hat{u}^{I}_{ps}|,|\check{u}^{I}_{ps}|\}\), \(u^{J}_{ps}=\max \{|\hat{u}^{J}_{ps}|,|\check{u}^{J}_{ps}|\}\), and \(u^{K}_{ps}=\max \{|\hat{u}^{K}_{ps}|,|\check{u}^{K}_{ps}|\}\).
Proof
From (16) is equivalent to prove
The proof of (16) will be splitted into the following two cases.
Case 1. \(y_1,y_2 \overline{\in }\;\; \Upsilon ^{T}_{p}\). Then \(\lambda ^{R}_{ps}\big ( y_{1}\big )=\lambda ^{R}_{ps}\big ( y_{2}\big )\), hence
Case 2. \(\,y_1\in \Upsilon ^{T}_{p}\) and \(y_2\; \overline{\in }\; \Upsilon ^{T}_{p}\)\(\Big (or \Big )\;y_2\in \Upsilon ^{T}_{p}\) and \(y_1 \overline{\in }\; \Upsilon ^{T}_{p}\). The Proof of the case 2 can be dived into two subcases, which is followed by
Subcase (i) \(\,F^{R}_{s}\le h_{2}\). Since \(\tilde{f}^{R}_{s}\big ( F^{R}_{s},q_1,w_1,z_1 \big ) =\tilde{f}^{R}_{s}\big ( F^{R}_{s},q_2,w_2,z_2 \big )=0\), one has
Subcase (ii) \(-F^{R}_{s}\le h_{2}\). Since \(\tilde{f}^{R}_{s}\big ( -F^{R}_{s},q_1,w_1,z_1 \big ) =\tilde{f}^{R}_{s}\big ( -F^{R}_{s},q_2,w_2,z_2 \big )=0\), one has
From the above two cases, inequality (16) holds, thus the proof of (20) is finished. The rest of the proof of (17)–(19) are similar to the proof of (16). Hence, it is omitted here. \(\square \)
Corollary 2.13
Under Assumptions \([\mathcal {A}_{1}]\) and \([\mathcal {A}_{2}]\), for any \(y_1=h_1+iq_1+jw_1+kz_1,\;y_2=h_2+iq_2+jw_2+kz_2\in \mathbb {Q}\), there is \(\tilde{f}^{R}_{s}\big ( \pm F^{R}_{s},q_{s}, w_{s},z_{s}\big ) = \tilde{f}^{I}_{s}\big (h_{s}, \pm F^{I}_{s},w_{s},z_{s}\big ) =\tilde{f}^{J}_{s}\big (h_{s}, q_{s},\pm F^{J}_{s},z_{s}\big )= \tilde{f}^{K}_{s}\big (h_{s}, q_{s},w_{s},\pm F^{K}_{s}\big )= 0\), then
where \(p,s\in \{1,2,\ldots ,m\}\), \(u^{R}_{ps}\), \(u^{I}_{ps}\), \(u^{J}_{ps}\), and \(u^{K}_{ps}\) are already defined in Lemma 2.12.
Considering \(y'_{p}(t)=\big ( y'_{1}(t),\ldots ,y'_{m}(t) \big )^{T}\) and \(y''_{p}(t)=\big ( y''_{1}(t),\ldots ,y''_{m}(t)\big )^{T}\) are any two solutions of FQMNNs (2) with different initial condition \(y'_{p}(0)=y'_{0p}\) and \(y''_{p}(0)=y''_{0p}\) for \(p\in \{1,2,\ldots ,m\}\).
Let \(y'_{p}(t)=h'_p(t)+iq'_p(t)+jw'_p(t)+kz'_p(t)\) and \(y''_{p}(t)=h''_p(t)+iq''_p(t)+jw''_p(t)+kz''_p(t)\), then
almost everywhere \(t\ge t_{0}\) and
where \(\lambda ^{R}_{ps}\big ( \cdot \big ),\;\tilde{\lambda }^{R}_{ps} \big ( \cdot \big )\in \overline{co}\{u^{R}_{ps}\}\big ( \cdot \big )\), \(\lambda ^{I}_{ps}\big ( \cdot \big ),\;\tilde{\lambda }^{I}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{I}_{ps}\}\big ( \cdot \big )\), \(\lambda ^{J}_{ps}\big ( \cdot \big ),\;\tilde{\lambda }^{J}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{J}_{ps}\}\big ( \cdot \big )\) and \(\lambda ^{K}_{ps}\big ( \cdot \big ),\;\tilde{\lambda }^{K}_{ps}\big ( \cdot \big )\in \overline{co}\{u^{K}_{ps}\}\big ( \cdot \big )\).
Definition 2.14
[18] Let \(y'(t)=\big (y'_{1}(t),\ldots ,y'_{m}(t)\big )^{T}\) and \(y''(t)=\big (y''_{1}(t),\ldots ,y''_{m}(t)\big )^{T}\) are any solutions of FQMNNs (2) is said to be Mittag-Leffler stable in finite time with respecting to \(\{\rho ,\varrho ,T\}\), if there exist constants \(\rho>\varrho >0\) such that
when
where \(t_{0}\) is starting time.
Definition 2.15
Let \(y'(t)=\big (y'_{1}(t),\ldots ,y'_{m}(t)\big )^T\) and \(y''(t)=\big (y''_{1}(t),\ldots ,y''_{m}(t)\big )^T\) are any solutions of FQMNNs (2) is said to be asymptotically stable, if
Let \(x_{p}(t)=y'_{p}(t)-y''_{p}(t)\), where \(x_{p}(t)=x^{R}_{p}(t)+ix^{I}_{p}(t)+jx^{J}_{p}(t)+kx^{K}_{p}(t)\), that is \(x^{R}_{p}(t)=h'_{p}(t)-h''_{p}(t)\), \(x^{I}_{p}(t)=q'_{p}(t)-q''_{p}(t)\), \(x^{J}_{p}(t)=w'_{p}(t)-w''_{p}(t)\) and \(x^{K}_{p}(t)=z'_{p}(t)-z''_{p}(t)\), then the error system is
where
Remark 2.16
In FQMNNs (2), if \(y_{p}(t)=h_{p}(t)+iq_{p}(t)\), and all the coefficients of (2) are assumed to complex coefficients, at that point FQMNNs (2) will turn to impulsive finite-time stability of fractional order memristive neural networks in complex field; If all the coefficients of (2) are assumed to real coefficients, at that point FQMNNs (2) will turn to impulsive finite-time stability of fractional order memristive neural networks in real field.
3 Main Results
In this section, we will present the finite time stability results for FQMNNs (2) with fractional order \(0<\beta <1\) and \(1<\beta <2\).
3.1 Fractional-Order \(0<\beta <1\)
In this subsection, we studies some novel sufficient conditions to guarantee the finite time stability of the solutions of impulsive FQMNNs (2) by using Lyapunov function, Mittag-Leffler function and fractional-order differential inequalities.
Theorem 3.1
Under Assumptions \([\mathcal {A}_1]-[\mathcal {A}_{2}]\), FQMNNs (2) is finite-time Mittag-Leffler stable if the following relationship holds:
- (i)
There exists constants \(\theta _{p\tau }, \;\varepsilon _{p\tau },\;\eta _{p\tau },\;\sigma _{p\tau }\), the functions \(S^{R}_{p\tau }(\cdot ),\;S^{I}_{p\tau }(\cdot ), \;S^{J}_{p\tau }(\cdot )\) and \(S^{K}_{p\tau }(\cdot )\) satisfies
$$\begin{aligned} S^{R}_{p_\tau }(h'_{p}(t)-h''_{p}(t))= & {} -\,\theta _{p\tau }(h'_{p}(t) -h''_{p}(t)),\;0<\theta _{p\tau }<2,\\ S^{I}_{p_\tau }(q'_{p}(t)-q''_{p}(t))= & {} -\,\varepsilon _{p\tau }(q'_{p}(t) -q''_{p}(t)),\;0<\varepsilon _{p\tau }<2,\\ S^{J}_{p_\tau }(w'_{p}(t)-w''_{p}(t))= & {} -\,\eta _{p\tau }(w'_{p}(t) -w''_{p}(t)),\;0<\eta _{p\tau }<2,\\ S^{K}_{p_\tau }(z'_{p}(t)-z''_{p}(t))= & {} -\,\sigma _{p\tau }(z'_{p}(t) -z''_{p}(t)),\;0<\sigma _{p\tau }<2. \end{aligned}$$ - (ii)
If there are m positive constants \(\xi _{p},\;\zeta _{p}\;\gamma _{p},\;\alpha _{p},\;p=1,2,\ldots ,m\) such that
$$\begin{aligned} F_{1p}= & {} -\xi _{p}a_{p}+\sum _{s=1}^{m}u^{R}_{sp} \big [\xi _{s}\Phi _{p}^{R1} +\zeta _{s}\Phi _{p}^{I1}+\gamma _{s}\Phi _{p}^{J1} +\alpha _{s}\Phi _{s}^{K1}\big ]\\&+\,\sum _{s=1}^{m}u^{I}_{sp}\big [\xi _{s}\Phi _{p}^{I1} +\zeta _{s}\Phi _{p}^{R1} +\gamma _{s}\Phi _{p}^{K1}+\alpha _{p}\Phi _{s}^{J1}\big ]\\&+\,\sum _{s=1}^{m}u^{J}_{sp} \big [\xi _{s}\Phi _{p}^{J1}+\zeta _{s}\Phi _{p}^{K1}+\gamma _{s}\Phi _{p}^{R1} +\alpha _{s}\Phi _{s}^{I1}\big ]\\&+\,\sum _{s=1}^{m}u^{K}_{sp} \big [\xi _{s}\Phi _{p}^{K1} +\zeta _{s}\Phi _{p}^{J1}+\gamma _{s}\Phi _{p}^{I1} +\alpha _{p}\Phi _{s}^{R1}\big ]<0\\ F_{2p}= & {} -\zeta _{p}a_{p}+\sum _{s=1}^{m}u^{R}_{sp} \big [\xi _{s}\Phi _{p}^{R2} +\zeta _{s}\Phi _{p}^{I2}+\gamma _{s}\Phi _{p}^{J2} +\alpha _{s}\Phi _{s}^{K2}\big ]\\&+\,\sum _{s=1}^{m}u^{I}_{sp}\big [\xi _{s}\Phi _{p}^{I2} +\zeta _{s}\Phi _{p}^{R2}+\gamma _{s}\Phi _{p}^{K2} +\alpha _{p}\Phi _{s}^{J2}\big ]\\&+\,\sum _{s=1}^{m}u^{J}_{sp}\big [\xi _{s}\Phi _{p}^{J2} +\zeta _{s}\Phi _{p}^{K2}+\gamma _{s}\Phi _{p}^{R2} +\alpha _{s}\Phi _{s}^{I2}\big ]\\&+\,\sum _{s=1}^{m}u^{K}_{sp}\big [\xi _{s}\Phi _{p}^{K2} +\zeta _{s}\Phi _{p}^{J2}+\gamma _{s}\Phi _{p}^{I2} +\alpha _{p}\Phi _{s}^{R2}\big ]<0\\ F_{3p}= & {} -\gamma _{p}a_{p}+\sum _{s=1}^{m}u^{R}_{sp} \big [\xi _{s}\Phi _{p}^{R3} +\zeta _{s}\Phi _{p}^{I3}+\gamma _{s}\Phi _{p}^{J3} +\alpha _{s}\Phi _{s}^{K3}\big ]\\&+\,\sum _{s=1}^{m}u^{I}_{sp}\big [\xi _{s}\Phi _{p}^{I3} +\zeta _{s}\Phi _{p}^{R3}+\gamma _{s}\Phi _{p}^{K3} +\alpha _{p}\Phi _{s}^{J3}\big ]\\&+\,\sum _{s=1}^{m}u^{J}_{sp}\big [\xi _{s}\Phi _{p}^{J3} +\zeta _{s}\Phi _{p}^{K3}+\gamma _{s}\Phi _{p}^{R3} +\alpha _{s}\Phi _{s}^{I3}\big ]\\&+\,\sum _{s=1}^{m}u^{K}_{sp}\big [\xi _{s}\Phi _{p}^{K3} +\zeta _{s}\Phi _{p}^{J3}+\gamma _{s}\Phi _{p}^{I3} +\alpha _{p}\Phi _{s}^{R3}\big ]<0\\ F_{4p}= & {} -\,\alpha _{p}a_{p}+\sum _{s=1}^{m}u^{R}_{sp} \big [\xi _{s}\Phi _{p}^{R4} +\zeta _{s}\Phi _{p}^{I4}+\gamma _{s}\Phi _{p}^{J4} +\alpha _{s}\Phi _{s}^{K4}\big ]\\&+\,\sum _{s=1}^{m}u^{I}_{sp}\big [\xi _{s}\Phi _{p}^{I4} +\zeta _{s}\Phi _{p}^{R4} +\gamma _{s}\Phi _{p}^{K4} +\alpha _{p}\Phi _{s}^{J4}\big ]\\&+\,\sum _{s=1}^{m}u^{J}_{sp}\big [\xi _{s}\Phi _{p}^{J4} +\zeta _{s}\Phi _{p}^{K4}+\gamma _{s}\Phi _{p}^{R4} +\alpha _{s}\Phi _{s}^{I4}\big ]\\&+\,\sum _{s=1}^{m}u^{K}_{sp}\big [\xi _{s}\Phi _{p}^{K4} +\zeta _{s}\Phi _{p}^{J4} +\gamma _{s}\Phi _{p}^{I4} +\alpha _{p}\Phi _{s}^{R4}\big ]<0. \end{aligned}$$ - (iii)
Furthermore,
$$\begin{aligned} E_{\beta ,1}\big (-\varpi t^{\beta } \big )<\frac{\rho }{\varrho } \end{aligned}$$where \(-\varpi =\max _{1\le p\le m}\big \{F_{1p},F_{2p}, F_{3p},F_{4p}\big \}>0\).
Proof
Consider the following Lyapunov functional
where
Firstly, we consider the case of \(t=t_\tau ,\;\tau =1,2,\ldots \), from condition (i) of Theorem 3.1, one has
Secondly, we consider the case of \(t\ne t_\tau ,\;\tau =1,2,\ldots \). Taking the fractional-order time derivative of V(t) along the trajectories of (30)–(33) and, based on Lemma 2.7, one can get
By virtue of Assumption \([\mathcal {A}_{2}]\) and Lemma 2.12, one gets
Then, we have
Then by virtue of Lemma 2.9, it is easy to get
That is,
then
By using the definition of \(\Vert (\cdot )\Vert _{1}\) and from (39), it can get
where \(\chi =\max _{1\le p\le 4m}\big \{\delta _{p}\big \}\). According to condition (iii) in Theorem 3.1, it is easily to get,
By utilizing Definition 2.14 and inequality (40), the FQMNNs (2) is finite-time Mittag-Leffler stable with respect to \(\{\varrho ,\rho ,T\}\) if there exists \(\Vert y'(0)-y''(0)\Vert \le \varrho \) then it implies \(\Vert y'(t)-y''(t)\Vert \le \rho \). The proof is accomplished. \(\square \)
Now we prove that the existence, uniqueness and global Mittag-Leffler stability of equilibrium point for FQMNNs (2).
Theorem 3.2
Under Assumptions \([\mathcal {A}_1]-[\mathcal {A}_{2}]\) and let \(L_{p}(t)\equiv L_{p}\in \mathbb {Q}\), FQMNNs (2) admits a unique equilibrium point which is finite time Mittag-Leffler stable if the following relationship holds:
- (i)
If there exist a m positive constants \(\delta _{p},\;p=1,2,\ldots ,m\) such that the following conditions hold:
$$\begin{aligned}&\delta _{p}a_{p}-\sum ^{m}_{s=1}\delta _{s}\Big [ u^{R}_{sp} \big ( \Phi ^{R1}_{p} +\Phi ^{R2}_{p}+\Phi ^{R3}_{p}+\Phi ^{R4}_{p}\big ) + u^{I}_{sp} \big ( \Phi ^{I1}_{p}+\Phi ^{I2}_{p}+\Phi ^{I3}_{p}+\Phi ^{I4}_{p}\big )\nonumber \\&\quad +\, u^{J}_{sp} \big ( \Phi ^{J1}_{p}+\Phi ^{J2}_{p}+\Phi ^{J3}_{p} +\Phi ^{J4}_{p}\big ) + u^{K}_{sp} \big ( \Phi ^{K1}_{p}+\Phi ^{K2}_{p} +\Phi ^{K3}_{p}+\Phi ^{K4}_{p}\big ) \Big ]>0, \end{aligned}$$(41)$$\begin{aligned}&\delta _{p}a_{p}-\sum ^{m}_{s=1}\delta _{s}\Big [ u^{R}_{sp} \big ( \Phi ^{I1}_{p}+\Phi ^{I2}_{p}+\Phi ^{I3}_{p}+\Phi ^{I4}_{p}\big )+ u^{I}_{sp} \big ( \Phi ^{R1}_{p}+\Phi ^{R2}_{p}+\Phi ^{R3}_{p}+\Phi ^{R4}_{p}\big )\nonumber \\&\quad +\, u^{J}_{sp} \big (\Phi ^{K1}_{p}+\Phi ^{K2}_{p}+\Phi ^{K3}_{p} +\Phi ^{K4}_{p} \big ) + u^{K}_{sp} \big ( \Phi ^{J1}_{p} +\Phi ^{J2}_{p}+\Phi ^{J3}_{p}+\Phi ^{J4}_{p}\big ) \Big ]>0, \end{aligned}$$(42)$$\begin{aligned}&\delta _{p}a_{p}-\sum ^{m}_{s=1}\delta _{s}\Big [ u^{R}_{sp} \big ( \Phi ^{J1}_{p}+\Phi ^{J2}_{p}+\Phi ^{J3}_{p}+\Phi ^{J4}_{p}\big )+ u^{I}_{sp} \big ( \Phi ^{K1}_{p}+\Phi ^{K2}_{p}+\Phi ^{K3}_{p}+\Phi ^{K4}_{p}\big )\nonumber \\&\quad +\, u^{J}_{sp} \big (\Phi ^{R1}_{p}+\Phi ^{R2}_{p}+\Phi ^{R3}_{p}+\Phi ^{R4}_{p} \big ) + u^{K}_{sp} \big ( \Phi ^{I1}_{p}+\Phi ^{I2}_{p}+\Phi ^{I3}_{p} +\Phi ^{I4}_{p}\big ) \Big ]>0, \end{aligned}$$(43)$$\begin{aligned}&\delta _{p}a_{p}-\sum ^{m}_{s=1}\delta _{s}\Big [ u^{R}_{sp} \big ( \Phi ^{K1}_{p}+\Phi ^{K2}_{p}+\Phi ^{K3}_{p}+\Phi ^{K4}_{p}\big )+ u^{I}_{sp} \big (\Phi ^{J1}_{p}+\Phi ^{J2}_{p}+\Phi ^{J3}_{p}+\Phi ^{J4}_{p}\big )\nonumber \\&\quad +\, u^{J}_{sp} \big (\Phi ^{I1}_{p}+\Phi ^{I2}_{p}+\Phi ^{I3}_{p}+\Phi ^{I4}_{p} \big ) + u^{K}_{sp} \big ( \Phi ^{R1}_{p}+\Phi ^{R2}_{p}+\Phi ^{R3}_{p} +\Phi ^{R4}_{p}\big ) \Big ]>0. \end{aligned}$$(44)
Proof
Given any \(\lambda _{ps}(\cdot )\in \overline{co}\{u_{ps}\}(\cdot )\), construct a contraction mapping \(\Lambda :\mathbb {Q}^{m}\rightarrow \mathbb {Q}^{m}\), \(\Lambda (y)=\big ( \Lambda _1(y),\ldots ,\Lambda _m(y) \big )^T\) and
Then
For any two quaternion vectors \(v_{1}=\big (v_{11},v_{12},\ldots ,v_{1m} \big )^{T}\), \(v_{2}=\big (v_{21},v_{22},\ldots ,v_{2m} \big )^{T}\in \mathbb {Q}^{m}\). Let \(v_{1s}=h_{1s}+iq_{1s}+jw_{1s}+kz_{1s}\), \(v_{2s}=h_{2s}+iq_{2s}+jw_{2s}+kz_{2s}\in \mathbb {Q}\). By virtue of Lemma 2.12, one has
By using Definition of \(\Vert \cdot \Vert _1\), and from (50)–(53), we have
Let
For any \(y=\big ( y_1,\ldots ,y_m \big )^T\in \mathbb {Q}^{m}\), let \(\Vert y\Vert _{1}=\sum ^{4m}_{p=1}|y_{p}|=\sum ^{m}_{p=1}\big [|y^{R}_{p}| +|y^{I}_{p}|+|y^{J}_{p}|+|y^{K}_{p}|\big ]\). From (47), one has
where \(\mu =\max \big \{ \max _{1\le p\le m}\{\mu ^{R}_{p}\},\max _{1\le p\le m}\{\mu ^{I}_{p}\}, \max _{1\le p\le m}\{\mu ^{J}_{p}\},\max _{1\le p\le m}\{\mu ^{K}_{p}\} \big \}\). From (34)–(37), it follows that \(\mu \in (0,1)\). According to (55), the mapping \(\Lambda :\mathbb {Q}^m\rightarrow \mathbb {Q}^m\) is a contraction mapping on \(\mathbb {Q}^{m}\). Therefore there exist a unique fixed point \(\Lambda (v^{\star })=v^{\star }\), i.e.,
Let \(y_{p}^{\star }=\frac{v^{\star }_p}{a_p\delta _{p}}\), it can obtain
that is
for \(p=1,2,\ldots ,m\), which implies that \(y^{\star }\) is an unique solution of FQMNNs (2). In addition, condition (41)–(44) imply conditions (ii) of Theorem 3.1 holds. By virtue of Theorem 3.1, if there exist a constant \(\varpi \) such that
Therefore, unique solution of FQMNNs (2) is finite-time Mittag-Leffler stable with respect to \(\{\varrho ,\rho ,T\}\) if there exists \(\Vert y(0)-y^{\star }\Vert \le \varrho \) then it implies \(\Vert y(t)-y^{\star }\Vert \le \rho \), and the proof of Theorem is completed. \(\square \)
Remark 3.3
When \(\beta =1\), model (2) degenerates into integer order finite time stability of QVMNNs.
Remark 3.4
When the state vector \(y_{p}(t)\), the memristive connection weight \(u_{ps}(y_s(t))\), the nonlinear activation \(f_{s}(y_s(t))\) and external input \(L_{p}(t)\) are all in complex domain (or real domain), model (2) can be reduced into finite time stability of complex (or real) valued memristive neural networks. So proposed in this model is more advanced.
Remark 3.5
The impulsive FQMNNs (2) is the corresponding closed-loop system to the control system
where \(\Delta _{p}(t)=\sum ^{+\infty }_{\tau =1} \delta (t-t_\tau ),\;p\in \{1,2,\ldots ,m\}\) is the control inputs, \(\delta (t)\) is the Dirac impulsive function. The impulsive controller has an effect on abrupt change of the states of (2) at \(t=t_\tau \) due to which the states of units change from the position \(y_{p}(t^{-}_\tau )\) into the position \(y_{p}(t^{+}_\tau )\), the function \(S_{p\tau }\) characterize the magnitudes of the impulse effects on the units \(y_{p}\) at \(t=t_\tau \). i.e., \(\Delta _{p}(t)\) is an impulsive controller of the FQMNNs
Therefore, Theorem 3.1 represents the general method of impulsive control law (3.3) for FQMNNs (57). The constants \(\theta _{p\tau }\), \(\varepsilon _{p\tau }\), \(\eta _{p\tau }\) and \(\sigma _{p\tau }\) in condition (i) in Theorem 3.1 characterize the stabilizing impulses. Therefore, our obtained results in Theorem 3.1 can be used to design impulsive control law under the controlled FQMNNs (2) are finite time stabilized onto FQMNNs (57).
3.2 Fractional-order \(1<\beta <2\)
When the impulses are not taken into consideration, FQMNNs (2) degenerates into the following expression:
or equivalently,
where \(A=diag\{a_1,\ldots ,a_m\}\), \(y(t)=\big ( y_{1}(t),\ldots ,y_{m}(t) \big )^T\), \(U(y(t))=\big (u_{ps}(y_{p}(t))\big )_{n\times n}\), \(f(y(t))=\big ( y_1(t),\ldots ,y_{m}(t) \big )^{T}\), \(L(t)=(L_1(t),\ldots ,L_{m}(t))^{T}\).
For the sake of convenience, we define \(U^{R} =\big (u^{R}_{ps} \big )_{m\times m}\), \(U^{I}=\big ( u^{I}_{ps} \big )_{m\times m}\), \(U^{J}=\big ( u^{J}_{ps} \big )_{m\times m}\), \(U^{K}=\big ( u^{K}_{ps} \big )_{m\times m}\), \(\Phi ^{R1}=\max _{1\le s\le m}\{\Phi _{s}^{R1}\}\), \(\Phi ^{R2}=\max _{1\le s\le m}\{\Phi _{s}^{R2}\}\), \(\Phi ^{R3}=\max _{1\le s\le m}\{\Phi _{s}^{R3}\}\), \(\Phi ^{R4}=\max _{1\le s\le m}\{\Phi _{s}^{R4}\}\), \(\Phi ^{I1}=\max _{1\le s\le m} \{\Phi _{s}^{I1}\}\), \(\Phi ^{I2}=\max _{1\le s\le m}\{\Phi _{s}^{I2}\}\), \(\Phi ^{I3}=\max _{1\le s\le m}\{\Phi _{s}^{I3}\}\), \(\Phi ^{I4}=\max _{1\le s\le m}\{\Phi _{s}^{I4}\}\), \(\Phi ^{J1}=\max _{1\le s\le m}\{\Phi _{s}^{J1}\}\), \(\Phi ^{J2}=\max _{1\le s\le m}\{\Phi _{s}^{J2}\}\), \(\Phi ^{J3}=\max _{1\le s\le m}\{\Phi _{s}^{J3}\}\), \(\Phi ^{J4}=\max _{1\le s\le m}\{\Phi _{s}^{J4}\}\), \(\Phi ^{K1}=\max _{1\le s\le m}\{\Phi _{s}^{K1}\}\), \(\Phi ^{K2}=\max _{1\le s\le m}\{\Phi _{s}^{K2}\}\), \(\Phi ^{K3}=\max _{1\le s\le m}\{\Phi _{s}^{K3}\}\) and \(\Phi ^{K4}=\max _{1\le s\le m}\{\Phi _{s}^{K4}\}\).
Then, we state the finite time stability results of FQMNNs (59) as the following theorem.
Theorem 3.6
Under Assumptions \([\mathcal {A}_1]-[\mathcal {A}_{2}]\), FQMNNs (59) is finite-time stable if the following relationship holds:
where \(\theta =\min _{1\le p\le m}\{a_{p}\}\), \(\Upsilon (t)=\max \{g_{1}(t),g_{2}(t),g_{3}(t),g_{4}(t)\}\),
Proof
Let \(y'(t)=h'(t)+iq'(t)+jw'(t)+kz'(t)\) and \(y''(t)=h''(t)+iq''(t)+jw''(t)+kz''(t)\) are any two solutions of FQMNNs (59) with initial values \(y'(0)=y'_0,\;y''(0)=y''_0\) and \(x(t)=y''(t)-y'(t)\), \(\eta (0)=y'(0)-y''(0)\), it can be obtained that
where \(\Psi ^{RR}\big ( \cdot \big )\), \(\Psi ^{II}\big ( \cdot \big )\), \(\Psi ^{JJ}\big ( \cdot \big )\Psi ^{KK}\big ( \cdot \big )\), \(\Psi ^{RI}\big ( \cdot \big )\), \(\Psi ^{IR}\big ( \cdot \big )\), \(\Psi ^{JK}\big ( \cdot \big )\), \(\Psi ^{KJ}\big ( \cdot \big )\), \(\Psi ^{RJ}\big ( \cdot \big )\), \(\Psi ^{IK}\big ( \cdot \big )\), \(\Psi ^{JR}\big ( \cdot \big )\), \(\Psi ^{KI}\big ( \cdot \big )\), \(\Psi ^{RK}\big ( \cdot \big )\), \(\Psi ^{IJ}\big ( \cdot \big )\), \(\Psi ^{JI}\big ( \cdot \big )\) and \(\Psi ^{KR}\big ( \cdot \big )\) are already defined in Sect. 2.
By virtue of Lemma 2.12 and, from \([\mathcal {A}_1]-[\mathcal {A}_{2}]\), it can be followed that
Use Laplace transform and inverse Laplace transform of both sides of (61)–(64), one has
By application of Lemma 2.8, and from (65), we have
Similarly,
\(\Vert x^{R}(t)\Vert +\Vert x^{I}(t)\Vert +\Vert x^{J}(t)\Vert +\Vert x^{K}(t)\Vert \)
We denote \(g_{\tau }(t)=\Vert U^{R}\Vert \Big (\Phi ^{R\tau } +\Phi ^{I\tau }+\Phi ^{J\tau }+\Phi ^{K\tau } \Big ) +\Vert U^{I}\Vert \Big (\Phi ^{I\tau }+\Phi ^{R\tau }+\Phi ^{K\tau }+\Phi ^{J\tau }\Big ) +\Vert U^{J}\Vert \Big (\Phi ^{J\tau }+\Phi ^{K\tau }+\Phi ^{R\tau }+\Phi ^{I\tau } \Big ) +\Vert U^{K}\Vert \Big (\Phi ^{K\tau }+\Phi ^{J\tau }+\Phi ^{R\tau } +\Phi ^{I\tau }\Big ),\;\tau =1,2,3,4,\; \varepsilon (t) =\Big [\big \Vert \eta ^{R}_{0}(0)\big \Vert +\big \Vert \eta ^{I}_{0}(0)\big \Vert +\big \Vert \eta ^{J}_{0}(0)\big \Vert +\big \Vert \eta ^{K}_{0}(0)\big \Vert \Big ] +t\big [\big \Vert \eta ^{R}_{1}(0)\big \Vert +\big \Vert \eta ^{I}_{1}(0)\big \Vert +\big \Vert \eta ^{J}_{1}(0)\big \Vert +\big \Vert \eta ^{K}_{1}(0)\big \Vert \Big ] =\Vert \eta _{0}(0)\Vert +t\Vert \eta _{1}(0)\Vert \), from inequality (73), we have
In light of famous generalized Gronwall Lemma 2.6, it follows that
Let \(\Vert \eta (0)\Vert <\varrho \), then we have
Hence, if the condition of Theorem 3.5 hold, we gain \(\Vert x(t)\Vert<\rho ,\;0\le t<T\), where \(\Upsilon (t)=\max _{1\le \tau \le 4}\{g_{\tau }(t)\}\). It means FQMNNs (59) is finite-time stable via Definition 2.14. Proof completed. \(\square \)
Corollary 3.7
Under Assumptions \([\mathcal {A}_1]-[\mathcal {A}_{2}]\), FQMNNs (59) is asymptotically stable if the following relationship holds:
where \(\Upsilon (t)\) and \(\theta \) both respectively, are defined in Theorem 3.5.
Proof
According to the proof Theorem 3.5, from (74), we can get
By means of Lemma 2.5, there exists two positive constants \(\Lambda _1,\Lambda _2>0\) such that
That is
Hence, if the condition of Corollary 3.6 hold, from (75), \(x(t)\rightarrow 0\), as \(t\rightarrow +\infty \), which implies FQMNNs (59) is asymptotically stable based on Definition 2.15. Proof completed. \(\square \)
Remark 3.8
If the memristive connection weights of FQMNNs (2) is invariable,i.e., \(u_{ps}\big ( y_{s}(t)\big )\) is constant, then FQMNNs (2) degenerates into finite-time stability of fractional order quaternion-valued neural networks.
4 Numerical Examples
To verify the advantage of the above theoretical results, two numerical computer simulations are performed in the following few lines.
Example 4.1
Consider the following two dimensional FQMNNs:
for \(p=1,2,\;t\ge 0\), where \(\beta =0.91\), \(\theta _{1\tau }=\theta _{2\tau }=1.7\), \(\varepsilon _{1\tau }=\varepsilon _{2\tau }=1.65\), \(\eta _{1\tau }=\eta _{2\tau }=1.5\), \(\sigma _{1\tau }=\sigma _{2\tau }=1.65\), \(a_{1}=a_{2}=6\), \(L_{1}(t)=L_{2}(t)=0\), the activation function \(f_{s} \big (y_{s}\big )=\tanh ( y_{s}),\;s=1,2\), and the memristive connection weights as follows:
here
It is obvious that the assumption \([\mathcal {A}_{1}]-[\mathcal {A}_{2}]\) holds with \(\Phi _{R\iota }=0.1\), \(\Phi _{I\iota }=0.15\), \(\Phi _{J\iota }=0.2\), \(\Phi _{K\iota }=0.25\) for \(\iota =1,2,3,4\). By Theorem 3.1, we select \(\xi _{p}=1,\;\zeta _{p}=1.5,\;\gamma _{p}=2,\;\alpha _{p} =1.5\) and by using of above values, and from conditions of Theorem 3.1, we have \(F_{11}=-1.605,\;F_{12}=-1.405\), \(F_{21}=-3.105,\;F_{22}=-2.905\), \(F_{31}=-4.605,\;F_{32}=-7.605\), \(F_{41}=-1.605,\;F_{42}=-7.405\). Obviously, the conditions of Theorem 3.1 holds, and t is replaced by T. Let from \(\rho =9,\;\varrho =0.9\), \(E_{\beta ,1}\big (-\varpi t^{\beta } \big )<\frac{\rho }{\varrho }\). Therefore, FQMNNs (76) is stable in finite time, that is \(T=0.7339\). From the Numerical simulations, the initial values of (76) are selected as: \(y(0) =\big (1.4-1.5i+1.5j-1.6k,-1.2+0.45i-1.2j+k \big )^{T}\). In Figs. 1, 2, 3 and 4 presents the state trajectories of a real part h(t) and three imaginary parts \(q(t),\;w(t),\;z(t)\) of system (76), which confirms the validity of Theorem 3.1.
The state trajectory of real parts \(y_{1}^{R}(t),\;y_{2}^{R}(t)\)
The state trajectory of imaginary parts \(y_{1}^{I}(t),\;y_{2}^{I}(t)\)
The state trajectory of imaginary parts \(y_{1}^{J}(t),\;y_{2}^{J}(t)\)
The state trajectory of imaginary parts \(y_{1}^{K}(t),\;y_{2}^{K}(t)\)
Example 4.2
Consider the following two dimensional FQMNNs:
The state trajectory of real parts \(y_{1}^{R}(t),\;y_{2}^{R}(t)\)
The state trajectory of imaginary parts \(y_{1}^{I}(t),\;y_{2}^{I}(t)\)
for \(p=1,2,\;t\ge 0\), where \(\beta =1.75\), \(a_{1}=a_{2}=8\), \(L_{1}(t)=L_{2}(t)=0\), the activation function \(f_{s} \big (y_{s}\big )=\tanh ( y_{s}),\;s=1,2\), and the memristive connection weights as follows:
here
It is obvious that the assumption \([\mathcal {A}_{1}]-[\mathcal {A}_{2}]\) holds with \(\Phi _{R1}=0.1\), \(\Phi _{R2}=0.05\), \(\Phi _{R3}=0.2\), \(\Phi _{R4}=0.15\), \(\Phi _{I\iota }=0.2\), \(\Phi _{J1}=0.3\), \(\Phi _{J2}=0.4\), \(\Phi _{J3}=0.1=\Phi _{J4}=0.25\), \(\Phi _{K\iota }=0.35\) for \(\iota =1,2,3,4\). The initial values of FQMNNs (76) are chosen as: \(y(0)=\big ( -1.5+0.7i+1.5j+2.4k,1.3-0.8i+2.4j-2.5k \big )^{T}\). Now, \(\Vert U^{R}\Vert =\Vert U^{I}\Vert =0.75\), \(\Vert U^{J}\Vert =0.85\), \(\Vert U^{K}\Vert =1.45\), \(\theta =6\), \(\Gamma (1.75)=0.919\), then \(\Upsilon (t)\Gamma (1.75)=5.3302\). Let \(\varrho =0.5\) and \(\rho =5\), from
Therefore, FQMNNs (77) is stable in finite time, that is \(T=1.984\). Furthermore,
then FQMNNs (77) is globally asymptotically stable based on Corollary 3.6. In Figs. 5, 6, 7 and 8 depicts the state curves of a real part h(t) and three imaginary parts \(q(t),\;w(t),\;z(t)\) of system (77), which also assure the effectiveness of Theorem 3.5.
The state trajectory of imaginary parts \(y_{1}^{J}(t),\;y_{2}^{J}(t)\)
The state trajectory of imaginary parts \(y_{1}^{K}(t),\;y_{2}^{K}(t)\)
5 Conclusion
Hereof, the finite time stability of fractional order quaternion-valued memristor-based neural networks with order \(0<\beta <1\) and \(1<\beta <2\) has been studied. First of all, A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed. Secondly, via differential inclusion theory, Filippov’s solution, contraction mapping principle, Lyapunov function, and concept of quaternion algebra, the existence and finite time Mittag-Leffler stability criteria of FQMNNs with impulses have obtained when fractional order satisfying \(0<\beta <1\). Moreover, when impulsive effects are not taken into consideration, the finite-time stability and asymptotic stability conditions of FQMNNs with order \(1<\beta <2\) was established with the aid of Mittag-Leffler function and the generalized Gronwall-inequality. Finally, we provide two numerical simulations to illustrate the correctness of the proposed main consequences.
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Pratap, A., Raja, R., Alzabut, J. et al. Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses. Neural Process Lett 51, 1485–1526 (2020). https://doi.org/10.1007/s11063-019-10154-1
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DOI: https://doi.org/10.1007/s11063-019-10154-1