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LMI conditions for some dynamical behaviors of fractionalorder quaternionvalued neural networks
Advances in Difference Equations volume 2019, Article number: 266 (2019)
Abstract
This paper addresses the issue of three dynamical behaviors including global MittagLeffler stability, robust stability and projection synchronization for fractionalorder quaternionvalued neural networks (FQVNNs). Some linear matrix inequality conditions for these dynamical behaviors of FQVNNs are given by Lyapunov stability theory, quaternion matrix theory, Homeomorphic mapping theory and fractional differential equation theory. Furthermore, these obtained sufficient conditions for stability and synchronization are superior to those in existing literature. Finally, three examples are given to illustrate the effectiveness of the theoretical results.
Introduction
Fractional calculus, as a prolongation of integer calculus, was traceable in the 17th century [1, 2]. In the past, fractional calculus have not been paid enough attention due to the lack of proper application background. In recent decades, various dynamic systems in many areas, such as fluid mechanics, diffusion and wave propagation, boundary layer effects and electromagnetic waves, can be modeled by fractionalorder equations, which are a wonderful means to describe the memory and genetic properties of sundry materials and processes compared with integerorder ones [3,4,5,6,7,8,9,10]. On account of this, the research of fractional calculus has fascinated the interest of many scholars in science and engineering [11, 12]. In recent years, fractional derivatives have been brought into neural networks, in which fractionalorder equations can describe their behaviors [13,14,15]. Thereafter, the dynamics of fractionalorder neural networks (FNNs) has been a topic of attention in control and system engineering.
It is worth mentioning that the qualitative analysis of the dynamical behaviors is a critical step for the designed FNNs in many real applications. Generally, the design value of the system parameters deviates from its actual value because of the influence caused by the inaccurate measurement and environmental factors. Accordingly, it is necessary to research the robust stability problem of the system. This problem requires us to set some appropriate intervals so that the system is always stable provided the parameters lie in these intervals. Furthermore, for drive–response FNNs, it is critical to design some kinds of control laws in order to ensure the synchronization between the drive and response ones. Recently, a great deal of outstanding results about the stability, the robust stability and synchronization of FNNs were obtained [16,17,18,19,20,21,22,23,24,25,26,27,28,29]. In [16], αstability and αsynchronization were investigated for FNNs. The finitetime stability is was investigated for FNNs in [17]. The robust stability problems for discretetime uncertain neural networks were discussed in [19]. The MittagLeffler stability and synchronization of FNNs with leakage timevarying delay were concerned in [20, 21], and the projective synchronization of FNNs was studied in [22, 23].
It should be noted that quaternion, discovered by British mathematician W.R. Hamilton in 1843 [30], provides a concise mathematical method to represent automorphisms in threedimensional and fourdimensional spaces. Compared with matrix representation, the quaternion one is more compact and the calculating speed is faster [31]. Therefore, quaternionbased applications are increasingly emerging in quantum mechanics, signal processing, computer graphics, orbital mechanics and other fields [32, 33]. Particularly, quaternionvalued neural networks (QVNNs), as a universal promotion of realvalued neural networks (RVNNs) and complexvalued neural networks (CVNNs), have been designed for digital image associative memories in [34]. In this application, the three imaginary parts ı, ȷ and κ of the designed QVNNs are employed to represent the three basic colors separately. In this way, the dimension of the system is greatly reduced and the computational efficiency is greatly improved. In recent years, many researchers have investigated multifarious dynamical behaviors of QVNNs [35,36,37,38,39,40,41,42,43]. The global μstability criteria for QVNNs with unbounded timevarying delays were established in [35, 36, 39]. The robust stability problem of QVNNs with time delays and parameter uncertainties was studied in [40]. The global exponential stability for QVNNs with timevarying delays was researched in [41]. In [42], the authors considered the dissipativity of QVNNs and obtained some succinct criteria for ensuring the QVNNs to be globally dissipative. In [43], a stability analysis was made for continuoustime and discretetime QVNNs with linear threshold neurons.
To the best of our knowledge, the existing literature on dynamical behaviors of fractionalorder QVNNs (FQVNNs) is very little [44]. Motivated by the above discussions, this paper will mainly focus on the global MittagLeffler stability, robust stability and projection synchronization of FQVNNs. The main contributions made in this paper are as follows:

(1)
Different from the approaches in the existing literature, we investigate the dynamical behaviors of FQVNNs directly instead of converting them into complexvalued or realvalued system, which avoids the increase of system dimension.

(2)
Inspired by the product rule for integerorder derivatives, the fractionalorder version of the rule is formulated by an inequality, which plays an important role in the computation of the fractionalorder derivative of Lyapunov functions in Lyapunov’s second method for analyzing dynamical behaviors of fractionalorder system.

(3)
The quaternionvalued linear matrix inequality (LMI) conditions for the dynamical behaviors of FQVNNs are converted into the complexvalued LMI ones, which can be tested directly by the mathematical software MATLAB.
Notation: In this paper, \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\) denote separately the real field, the complex field and the skew field of a quaternion. \(\mathbb{R}^{n\times m}\), \(\mathbb{C}^{n \times m}\) and \(\mathbb{H}^{n\times m}\), simply, \(\mathbb{R}^{n}\), \(\mathbb{C}^{n}\), and \(\mathbb{H}^{n}\) when \(m=1\), denote separately \(n\times m\) matrices with entries from \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\). The standard imaginary units in \(\mathbb{H}\) are denoted by ı, ȷ and κ which satisfy \(\imath ^{2}=\jmath ^{2}=\kappa ^{2}=1\), \(\imath \jmath =\jmath \imath =\kappa \), \(\jmath \kappa =\kappa \jmath =\imath \), \(\kappa \imath =\imath \kappa =\jmath \). For a quaternion \(q=q_{0}+q_{1}\imath +q_{2}\jmath +q _{3}\kappa =(q_{0}+q_{1}\imath )+(q_{2}+q_{3}\imath )\jmath \in \mathbb{H}\), where \(q_{0},q_{1},q_{2},q_{3}\in \mathbb{R}\). Let \(q^{*}=q_{0}q_{1}\imath q _{2}\jmath q_{3}\kappa \) be the conjugate of q, and \(q=\sqrt{q_{0} ^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}\) be the modulus of q. For \(z=(z_{1},z_{2},\ldots ,z_{n})^{T}\in \mathbb{H}^{n}\), let \(z=(z _{1},z_{2},\ldots ,z_{n})^{T}\) be the modulus of z, and \(\z\=\sqrt{\sum_{i=1}^{n}z_{i}^{2}}\) be the norm of z. In addition, let \(h^{(11)}, h^{(12)}, h^{(21)}, h^{(22)}\in \mathbb{R}\) denote separately the first, second, third and fourth parts of h, that is, \(h^{(11)}=q_{0}\), \(h^{(12)}=q_{1}\), \(h^{(21)}=q_{2}\) and \(h^{(22)}=q_{3}\), and let \(h_{1}, h_{2}\in \mathbb{C}\) denote separately the first and second complex parts of h, that is, \(h_{1}=q_{0}+q _{1}\imath \) and \(h_{2}=q_{2}+q_{3}\imath \). In the same way, for \(A\in \mathbb{H}^{n\times m}\), the first and second complex parts are denoted by \(A_{1}\) and \(A_{2}\), and we have \(A=A_{1}+A_{2}\imath \). The notation Ā, \(A^{T}\) and \(A^{*}\) stand for the conjugate, the transpose and the conjugate transpose, separately, of the matrix A. For \(A=(a_{ij})_{n\times n}\in \mathbb{H}^{n\times n}\), let \(\A\=\sqrt{\sum_{i=1}^{n}\sum_{j=1}^{n} a_{ij}^{2}}\) denote the norm of A. I denotes the identity matrix with appropriate dimensions. The notation \(X\geq Y\) (separately, \(X>Y\)) means that \(XY\) is positive semidefinite (separately, positive definite). For a positive definite Hermitian matrix \(P\in \mathbb{H}^{n\times n}\), \(\lambda _{\max }(P)\) and \(\lambda _{\min }(P)\) are defined as the largest and the smallest eigenvalues of P, separately.
Model description and preliminaries
In order to describe the model considered in this paper, we first introduce the definition of the fractional derivative.
Definition 1
([22])
The Caputo fractional derivative of order α for a function \(f(t)\in C^{n+1}([0,+\infty ),\mathbb{H})\) is defined as
where \(\alpha >0\), \(\varGamma (\cdot )\) represents the gamma function and n is a positive integer such that \(n1<\alpha <n\). Particularly, when \(0<\alpha <1\),
The Laplace transform of the Caputo fractionalorder derivative is
where \(n1<\alpha \leq n\), \(\mathcal{L}\{\cdot \}\) is the Laplace transform, s is the variable in the Laplace domain, and \(F(s)= \mathcal{L}\{f(t)\}\).
We consider the FQVNNs model with the following form:
where \(\alpha \in (0,1)\), \(t\geq 0\), \(x(t)=(x_{1}(t), x_{2}(t),\ldots, x _{n}(t))^{T}\in \mathbb{H}^{n}\) is the state vector of the neural network with n neurons at time t; \(C=\text{diag}\{c_{1},c_{2},\ldots,c _{n}\}\in \mathbb{R}^{n\times n}\) is the selffeedback connection weight matrix; \(B\in \mathbb{H}^{n\times n}\) is the interconnection weight matrix; \(f(x(t))=(f_{1}(x(t)), f_{2}(x(t)),\ldots, f_{n}(x(t)))^{T} \in \mathbb{H}^{n}\) denotes the neuron activation at time t, and \(I\in \mathbb{H}^{n}\) denotes the external input vector.
The initial condition associated with system (2) is of the form
where \(x_{0}\in \mathbb{H}^{n}\).
In order to study the dynamical behaviors of system (2), the Lipschitz condition for the activation function is usually needed. So we assume that
Assumption 1
For \(i=1,2,\ldots,n\), \(f_{i}(x)\) is continuous and satisfies
where \(l_{i}\) is a real constant. Moreover, define \(L= \operatorname{diag}\{l_{1},l_{2},\ldots,l_{n}\}\).
In addition, the following assumption on parameter ranges is needed to investigate the robust stability of system (2).
Assumption 2
The parameters C, B, I in FQVNNs (2) are assumed to be in the following sets, respectively:
where Č, Ĉ, B̌, \(\hat{B}\in \mathbb{H}^{n \times n}\) and Ǐ, \(\hat{I}\in \mathbb{H}^{n}\). Furthermore, let \(\check{C}=(\check{c}_{ij})_{n\times n}\), \(\hat{C}=(\hat{c}_{ij})_{n \times n}\), \(\check{B}=(\check{b}_{ij})_{n\times n}\), \(\hat{B}=( \hat{b}_{ij})_{n\times n}\). Then we define \(\tilde{C}=(\tilde{c}_{ij})_{n \times n}\), \(\tilde{B}=(\tilde{b}_{ij})_{n\times n}\), where \(\tilde{c}_{ij}=\max \{\check{c}_{ij},\hat{c}_{ij}\}\), \(\tilde{b} _{ij}=\max \{\check{b}_{ij},\hat{b}_{ij}\}\).
In order to discuss the synchronization problem between two FQVNNs, we introduce the response system associated with the drive system (2) as follows:
where \(y(t)\in \mathbb{H}^{n}\) is the state vectors, \(u(t)\in \mathbb{H}^{n}\) is a suitable control law to be designed, and other parameters are the same as (2).
Next we introduce some definitions and lemmas about fractionalorder system and quaternion matrix.
Definition 2
([22])
With two parameters \(\alpha >0\) and \(\beta >0\), the MittagLeffler function is defined as
When \(\beta =1\), its oneparameter form can be rewritten as
where \(z\in \mathbb{H}\). Particularly, \(E_{1}(z)=e^{z}\).
Definition 3
The constant \(\tilde{x}\in \mathbb{H}^{n}\) is an equilibrium point of FQVNNs (2) if and only if
Definition 4
An equilibrium point x̃ of FQVNNs (2) is said to be MittagLeffler stable if there exist two positive constants λ and μ, such that, for any solution \(x(t)\) of FQVNNs (2) with the initial condition (3), one has
where \(m(\cdot )\) is a locally Lipschitz function on \(\mathbb{D} \subseteq \mathbb{H}^{n}\) satisfying \(m(0)=0\) and \(m(x)\geq 0\).
Definition 5
FQVNNs (2) with the parameters ranges defined by Assumption 2 are globally MittagLeffler robust stable if the unique equilibrium point of (2) is MittagLeffler robust stable for all \(C\in C_{I}\), \(B\in B_{I}\) and \(I\in I_{I}\).
Definition 6
Systems (2) and (4) are globally MittagLeffler projective synchronized, if there exists a projective coefficient matrix \(\varUpsilon \in \mathbb{H}^{n\times n}\), two positive constants λ and μ such that, for any two solutions \(x(t)\) and \(y(t)\) of system (2) and system (4) with different initial values denoted by \(x_{0},y_{0}\in \mathbb{H}^{n}\), one has
where \(m(\cdot )\) is a locally Lipschitz function on \(\mathbb{D} \subseteq \mathbb{H}^{n}\) satisfying \(m(0)=0\) and \(m(x)\geq 0\).
Lemma 1
([45])
Let \(A=A_{1}+A_{2}\jmath \) and \(B=B_{1}+B_{2}\jmath \), where \(A_{1}, A_{2},B_{1}, B_{2}\in \mathbb{C}^{n\times n}\) and \(A, B\in \mathbb{H}^{n\times n}\). Then:

(1)
\(A^{*}=A_{1}^{*}A_{2}^{T}\jmath \);

(2)
\(AB=(A_{1}B_{1}A_{2}\overline{B}_{2})+(A_{1}B_{2}+A_{2} \overline{B}_{1})\jmath \), where \(\overline{B}_{1}\) and \(\overline{B} _{2}\) are the conjugate of \(B_{1}\) and \(B_{2}\), respectively.
Lemma 2
([45])
Let \(Q\in \mathbb{H}^{n\times n}\) be a Hermitian matrix and \(Q=Q_{1}+Q_{2}\jmath \), where \(Q_{1}, Q_{2}\in \mathbb{C}^{n\times n}\). Then \(Q<0\) is equivalent to
where \(\overline{Q}_{1}\), \(\overline{Q}_{2}\) are the conjugate matrices of \(Q_{1}\) and \(Q_{2}\), respectively.
Lemma 3
([40])
For any \(a,b\in \mathbb{H}^{n}\), if \(P\in \mathbb{H}^{n\times n}\) is a positive definite Hermitian matrix, then \(a^{*}b+b^{*}a\leq a^{*}Pa+b ^{*}P^{1}b\).
Lemma 4
([40])
A given real symmetric matrix
where \(S_{11}^{T}=S_{11}\), \(S_{12}^{T}=S_{21}\), and \(S_{22}^{T}=S_{22}\), if and only if any of the following two conditions holds:

(i)
\(S_{22}<0\) and \(S_{11}S_{12}S_{22}^{1}S_{21}<0\),

(ii)
\(S_{11}<0\) and \(S_{22}S_{21}S_{11}^{1}S_{12}<0\).
Lemma 5
([40])
If \(H(x):\mathbb{H}^{n}\rightarrow \mathbb{H}^{n}\) is a continuous map and satisfies the following conditions:

(i)
\(H(x)\) is injective on \(\mathbb{H}^{n}\),

(ii)
\(\lim_{\x\\rightarrow \infty }\H(x)\=\infty \).
Then \(H(x)\) is a homeomorphism of \(\mathbb{H}^{n}\) onto itself.
Lemma 6
Let x̃ is an equilibrium point of FQVNNs (2) with initial condition (3), and \(\mathbb{D}\subseteq \mathbb{H}^{n}\) be a domain containing the origin. Let \(V(x): \mathbb{D}\rightarrow \mathbb{R}\) be a continuously differentiable function and locally Lipschitz such that
where \(t\geq 0\), \(\alpha \in (0,1)\), \(x(t)\in \mathbb{D}\), \(a_{1}\), \(a_{2}\), \(a_{3}\), a, and b are arbitrary positive constants. Then x̃ is MittagLeffler stable.
Proof
By (6) and (7), it is obvious that
It follows from (8) and (9) that
where \(\beta =a_{2}^{1}a_{3}\). Then there exists \(M(t)\geq 0\) such that
By the application of Laplace transform to (10), we obtain
where \(V_{0}=V(x_{0}\tilde{x})\), \(\mathcal{V}(s)=\mathcal{L}\{V(x(t) \tilde{x})\}\), \(\mathcal{M}(s)=\mathcal{L}\{M(t)\}\), \(\mathcal{L}\{ \cdot \}\) is the Laplace transform. From (11), we could compute
On the one hand, when \(x_{0}=\tilde{x}\), we see that \(x(t)=\tilde{x}\) is the solution of (2) and \(V_{0}=0\). On the other hand, when \(x_{0}\neq \tilde{x}\), then \(V_{0}>0\). Since \(V(x)\) is locally Lipschitz, we get
by the inverse Laplace transform of (12). Noting that \(t^{\alpha 1}\geq 0\) and \(E_{\alpha ,\alpha }(\beta t^{\alpha })\geq 0\) for all \(t\in [0,+\infty )\), we have
In view of (8), it follows that
where \(m(x_{0}\tilde{x})=a_{1}^{1}V_{0}=a_{1}^{1}V(x_{0}\tilde{x})\). Obviously, \(m(x_{0}\tilde{x})\geq 0\) and \(m(x_{0}\tilde{x})=0\) if and only if \(x_{0}\tilde{x}=0\). Moreover, \(m(\cdot )\) is locally Lipschitz because \(V(\cdot )\) is locally Lipschitz. Therefore, the equilibrium point x̃ of FQVNNs (2) is MittagLeffler stable by Definition 4. □
Lemma 7
Let \(x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}\in \mathbb{H}^{n}\), where \(x_{i}(t),i=1,2,\ldots,n\), are continuous and differentiable function, and \(P\in \mathbb{H}^{n\times n}\) be a positive definite Hermitian matrix. Then, for \(t\geq 0\), the following inequality holds:
Proof
By Definition 1, we see that
Let \(y(\tau )=x(t)x(\tau )\). Then (14) can be written as
On the other hand, by using Lopida’s law, we could compute
Substituting (16) into (15), we obtain
since P is a positive definite Hermitian matrix. □
Lemma 8
([40])
Suppose \(A\in \mathbb{H}^{n\times n}\), \(\check{A}=(\check{a}_{ij})_{n \times n}\in \mathbb{H}^{n\times n}\), \(\hat{A}=(\hat{a}_{ij})_{n \times n}\in \mathbb{H}^{n\times n}\), and \(\check{A}\preceq A\preceq \hat{A}\). Then, for all \(x,y\in \mathbb{H}^{n}\), the following inequalities hold:
where \(\tilde{A}=(\tilde{a}_{ij})_{n\times n}\), \(\tilde{a}_{ij}= \max \{\check{a}_{ij},\hat{a}_{ij}\}\).
Remark 1
In Definition 6, we say system (2) and system (4) are globally MittagLeffler complete projective synchronized if \(\varUpsilon =E\), and they are globally MittagLeffler projective antisynchronized if \(\varUpsilon =E\).
Remark 2
The class of the activation functions \(f(x(t))\in \mathbb{H}^{n}\) satisfying Assumption 1 includes the linear threshold function defined in [44].
Remark 3
Differen from the product rule for integerorder derivatives, the fractionalorder version of the rule in Lemma 7 is presented by an inequality, which will greatly reduce the complexity of calculating the fractionalorder derivative of the Lyapunov functions in the proofs for the following main theorems.
Main results
In this section, we will discuss the existence, uniqueness, global MittagLeffler stability and global MittagLeffler robust stability of the equilibrium point FQVNNs (2) as well as the synchronization of FQVNNs (2).
Theorem 1
Under Assumption 1, FQVNNs (2) have a unique equilibrium point x̃ which is globally MittagLeffler stable, if there exist a real positive diagonal matrix Q and a Hermitian matrix \(P>0\) satisfying
Proof
First, we prove that system (2) has the unique equilibrium point by the homeomorphic mapping theory. Define a mapping \(\varTheta ( \omega ): \mathbb{H}^{n}\rightarrow \mathbb{H}^{n}\) and \(\varTheta ( \omega )=C\omega +Bf(\omega )+I\).
On the one hand, we prove the mapping \(\varTheta (\omega )\) is injective on \(\mathbb{H}^{n}\). If there exist \(\omega , \tilde{\omega }\in H ^{n}\) with \(\omega \neq \tilde{\omega }\) such that \(\varTheta (\omega )= \varTheta (\tilde{\omega })\). By Assumption 1, Lemma 3 and Lemma 8, we can compute
From Lemma 4 and the LMI condition (17) we obtain \(C^{*}PPC+PBQ^{1}B^{*}P+LQL<0\), which means that \(\omega =\tilde{\omega }\). Then \(\varTheta (\omega )\) is an injective mapping.
On the other hand, we need to prove \(\\tilde{\varTheta }(\omega )\ \rightarrow \infty \) as \(\\omega \\rightarrow \infty \). Let \(\tilde{\varTheta }(\omega )=\varTheta (\omega )\varTheta (0)\). According to Assumption 1, Lemma 3 and Lemma 8, we obtain
It follows from the Cauchy–Schwarz inequality that
When \(\omega \neq 0\), we have
Therefore, \(\\tilde{\varTheta }(\omega )\\rightarrow \infty \) as \(\\omega \\rightarrow \infty \). From above two aspects, we could find that \(\varTheta (\omega )\) is a homeomorphism on \(\mathbb{H}^{n}\) by Lemma 5. Hence, system (2) has a unique equilibrium point.
Then we prove the equilibrium point x̃ of system (2) is globally MittagLeffler stable.
With the variable substitution \(y(t)=x(t)\tilde{x}\), we could rewrite system (2) as
where \(g(y(t))=f(y(t)+\tilde{x})f(\tilde{x})\), \(g(0)=0\), and for all \(u,v\in \mathbb{H}\), \(g(x)\) satisfies
By doing this variable substitution, system (2) equals the system
It is obvious that \(\tilde{y}=0\) is the equilibrium point of system (20). Consider a Lyapunov function as follows:
It follows from (21) that
Thus, the condition (6) in Lemma 6 is satisfied. Then we do the following calculations by Assumption 1, Lemma 3 and Lemma 7:
Based on Lemma 4 and condition (17), we see that \(\lambda _{\max }(C^{*}PPC+PBQ ^{1}B^{*}P+LQL)<0\). It shows that the condition (7) in Lemma 6 is satisfied. Hence, the equilibrium point \(\tilde{y}=0\) of system (20) is globally MittagLeffler stable by Lemma 6. Namely, the equilibrium point x̃ of (2) is globally MittagLeffler stable. □
Theorem 2
Under Assumptions 1 and 2, FQVNNs (2) have a unique equilibrium point and the equilibrium point is globally MittagLeffler robust stable, if there exist a real positive diagonal matrix Q and a Hermitian matrices \(P>0\) such that the following LMI holds:
Proof
Similar to the proof of Theorem 1, we define a mapping \(\varTheta (\omega )=C\omega +Bf(\omega )+I\). First, we prove the mapping is injective. If there exist \(\omega ,\tilde{\omega }\in H ^{n}\) with \(\omega \neq \tilde{\omega }\) such that \(\varTheta (\omega )= \varTheta (\tilde{\omega })\). Based on (18) and Lemma 8, we could write
From Lemma 4 and the LMI condition (23), we have \(\check{C}^{*}PP\check{C}+P \tilde{B}Q^{1}\tilde{B}^{*}P+LQL<0\), which means \(\omega = \tilde{\omega }\). Hence \(\varTheta (\omega )\) is an injective mapping.
Let \(\tilde{\varTheta }(\omega )=\varTheta (\omega )\varTheta (0)\). According to (19) and Lemma 8 we get
It follows from the Cauchy–Schwarz inequality that
When \(\omega \neq 0\), we have
It is obvious that \(\\tilde{\varTheta }(\omega )\\rightarrow \infty \) as \(\\omega \\rightarrow \infty \). From Lemma 5, \(\varTheta (\omega )\) is a homeomorphism of \(\mathbb{H}^{n}\). Therefore, system (2) has the unique equilibrium point.
With the variable substitution \(y(t)=x(t)\tilde{x}\), then we could rewrite system (2) as
where \(g(y(t))=f(y(t)+\tilde{x})f(\tilde{x})\), \(g(0)=0\), and for all \(x,y\in \mathbb{H}\), \(g(x)\) satisfies
Construct a Lyapunov function as
Then the Lyapunov function satisfies
Hence, the Lyapunov function satisfies the condition (6). By (22) and Lemma 8 we could obtain
From Lemma 4 and condition (23) we could see that \(\lambda _{\max }( \check{C}^{*}PP\check{C}+P\tilde{B}Q^{1}\tilde{B}^{*}P+LQL)<0\). It shows that the condition (7) in Lemma 6 is satisfied. Thus, the equilibrium point \(\tilde{y}=0\) of (24) is globally MittagLeffler stable for all \(C\in C_{I}\), \(B\in B_{I}\), \(I\in I_{I}\) by Lemma 6 and Assumption 2. Namely, the equilibrium point x̃ of system (2) is globally MittagLeffler stable for all \(C\in C_{I}\), \(B\in B_{I}\), \(I\in I_{I}\). It follows from Definition 5 that system (2) is globally MittagLeffler robust stable. □
Before presenting the result on the synchronization between the drive system (2) and the response system (4), we should design the form of the control law \(u(t)\) in system (4). Let \(e(t)=y(t)\varUpsilon x(t)\), where \(\varUpsilon \in \mathbb{H}^{n\times n}\) a projective coefficient matrix. Then from (2) and (4) we get the error system as follows:
Choose the control law \(u(t)\) as
where \(K\in \mathbb{R}^{n\times n}\) is the coefficient matrix of the linear control \(w(t)\). In fact, the control scheme (25) is a hybrid control, \(v(t)\) is an open loop control, and \(w(t)\) is a linear control.
Then we could get the following error system by using the control law (25):
where \(h(e(t))=f(y(t))f(\varUpsilon x(t))\).
Theorem 3
System (2) and system (4) realize global projective synchronization under the control law (25), if Assumption (1) holds and there exists a real diagonal matrix Q and a Hermitian matrix \(P>0\), such that control coefficient matrix K satisfies
Proof
It is clear that \(\tilde{e}=0\) is an equilibrium point of system (26).
Construct a Lyapunov function as follows:
Then the following inequality holds:
which makes the condition (6) in Lemma 6 hold. From Assumption 1 and (26), we obtain
Then, according to Lemmas 3, 7 and (29), we do the following calculations:
Based on Lemma 4 and condition (17), we get \(\lambda _{\max }((KC)^{*}P+P(KC)+PBQ ^{1}B^{*}P+LQL)<0\). It shows that the condition (7) in Lemma 6 is satisfied. Hence the equilibrium point \(\tilde{e}=0\) of system (20) is globally MittagLeffler stable based on Lemma 6. Therefore, system (2) and system (4) are globally projective synchronized under the control law (25) by Definitions 4 and 6. □
Remark 4
In Theorems 1, 2 and 3, the LMI conditions for the global MittagLeffler stability, robust stability and projection synchronization of FQVNNs by Lyapunov stability theory were given, respectively. Unlike the methods in the existing literature [35, 41, 44], we considered FQVNNs directly instead of converting them into complexvalued or realvalued system.
Some corollaries
In this section, we will transform the quaternionvalued LMI conditions (17), (23) and (27) into complexvalued ones.
The quaternionvalued parameters A, B, and P could be expressed by the following complex forms: \(A=A_{1}+A_{2}\jmath \), \(B=B_{1}+B_{2} \jmath \) and \(P=P_{1}+P_{2}\jmath \), where \(A_{1}, A_{2}, B_{1}, B _{2}, P_{1}, P_{2}\in \mathbb{C}^{n\times n}\).
Corollary 1
Under Assumption 1, FQVNNs (2) have a unique equilibrium point x̃ which is globally MittagLeffler stable, if there exist a real positive diagonal matrix Q, a Hermitian matrix \(P_{1}\in \mathbb{C}^{n\times n}\) and a skewsymmetric matrix \(P_{2}\in \mathbb{C}^{n\times n}\) satisfying
where
Proof
By using Lemmas 1 and 2 as well as Theorem 1, the corollary can be proved straightforwardly. □
Corollary 2
Under Assumptions 1 and 2, FQVNNs (2) have a unique equilibrium point and the equilibrium point is globally MittagLeffler robust stable, if there exist a real positive diagonal matrix Q, a Hermitian matrix \(P_{1}\in C^{n\times n}\) and a skewsymmetric matrix \(P_{2}\in C^{n\times n}\) such that the following LMI holds:
where
Proof
By using Lemmas 1 and 2 as well as Theorem 2, the corollary can be proved straightforwardly. □
Corollary 3
System (2) and system (4) realize global projective synchronization under the control law (25), if Assumption 1 holds and there exist a real positive diagonal matrix Q, a Hermitian matrix \(P_{1}\in \mathbb{C} ^{n\times n}\), and a skewsymmetric matrix \(P_{2}\in \mathbb{C}^{n \times n}\), such that the control coefficient matrix K satisfies
where
with \(N_{11}=(KC)^{*}P_{1}+P_{1}(KC)+LQL\).
Proof
By using Lemmas 1 and 2 as well as Theorem 3, the corollary can be proved straightforwardly. □
Remark 5
The LMIs (17), (23) and (27) are quaternionvalued in Theorems 1, 2 and 3, separately, which cannot be processed directly through the MATLAB LMI toolbox. By using Lemmas 1 and 2, these quaternionvalued LMIs are transformed into the complexvalued ones in the above three corollaries, which can be checked easily by the mathematical software MATLAB.
Numerical examples
In this section, there are three examples to demonstrate the effectiveness of our results.
Example 1
Consider the following 2neuron FQVNNs:
where \(\alpha =0.95\), \(C=\text{diag}\{2,2\}\), \(f(x)=\sin (x_{0})+ \imath \sin (x_{1})+\jmath \sin (x_{2})+\kappa \sin (x_{3})\), \(I=(0.20.2\imath 0.3\jmath +0.4\kappa ,0.4+0.3\imath +0.4\jmath 0.3 \kappa )\), and \(B=(b_{ij})_{2\times 2}\), where
It is obvious that the parts of the parameter B are
By the YALMIP toolbox in MATLAB, we could find the following feasible solution to the LMIs (30):
Therefore, system (37) has a unique equilibrium point and the unique equilibrium point is globally MittagLeffler stable by Corollary 1. In the numerical simulation, the initial values are selected as \(x_{10}=2.5+1.5\imath +4.5\jmath +3.5\kappa \) and \(x_{20}=2.51.5\imath 4.5\jmath 3.5\kappa \). Figure 1 depicts the time responses of four parts of the state variable of (37), which validates the effectiveness of Corollary 1.
Remark 6
In [44], the global MittagLeffler stability problem for FQVNNs is considered. If we use Theorem 3 in [44] to test the existence and MittagLeffler stability of the equilibrium point of FQVNNs (37) in Example 1 of our article. We need check \(\hat{C}\hat{A}\), defined in [44], is a nonsingular Mmatrix. By the parameters of FQVNNs (37), we could find that \(\lambda (\hat{C} \hat{A})=0.3368\), 2.6985, 1.3715, 1.4318, 2.2030, 1.9982, 1.7668, 1.8670. Then \(\hat{C}\hat{A}\) is not a nonsingular Mmatrix. So we cannot check the existence and MittagLeffler stability of the equilibrium point of FQVNNs (37) in Example 1 by Theorem 3 in [44].
Example 2
Under Assumption 2, consider a 2neuron FQVNN as follows:
where
where
We could get the following matrix by matrices B̌ and B̂:
Letting \(L=\text{diag}\{0.3,0.3\}\), we could get the following feasible solutions of LMI (33) in Corollary 2 by suing the YALMIP toolbox in MATLAB:
Thus the conditions in Corollary 2 are satisfied. Then system (38) has a unique equilibrium point and the equilibrium point is globally robust stable.
Now we consider a special model of this example. We choose the following fixed network parameters: \(C=\text{diag}\{1,2\}\), \(B=(b_{ij})_{2 \times 2}\), \(f(x)=\sin (x_{0})+\imath \sin (x_{1})+\jmath \sin (x_{2})+ \kappa \sin (x_{3})\), and \(I=(0.20.2\imath 0.3\jmath +0.4\kappa ,0.4+0.3 \imath +0.4\jmath 0.3\kappa )\), where
By numerical simulation in MATLAB, we get the four parts of the states decided by the considered system, which initial conditions are chosen by 10 random constant quaternionvalued vectors. Moreover, it is showed in Fig. 2, which depicts that each neuron state converges to the stable state.
Example 3
Consider the following 2neuron FQVNNs as the drive system:
where \(\alpha =0.95\), \(C=\text{diag}\{1,1\}\), \(f(x)=\tanh (x_{0})+ \imath \tanh (x_{1})+\jmath \tanh (x_{2})+\kappa \tanh (x_{3})\), \(I=(0.20.2\imath 0.3\jmath +0.4\kappa ,0.4+0.3\imath +0.4\jmath 0.3 \kappa )\) and \(B=(b_{ij})_{2\times 2}\)
Then the corresponding response system is showed as follows:
where \(\alpha =0.95\), \(y(t)\in \mathbb{H}^{n}\), \(f(y)=\tanh (y_{0})+ \imath \tanh (y_{1})+\jmath \tanh (y_{2})+\kappa \tanh (y_{3})\), C, B and I are the same as that in (39), the control law \(u(t)\) in (40) is designed as (25). Therefore, the error system between drive system (39) and response system (40) is described by
where \(g(e(t))=f(y(t))\varUpsilon f(x(t))\).
We choose \(L=\operatorname{diag}\{0.3,0.3\}\), and
It is easy to get the parts of the parameter B:
By the YALMIP toolbox in MATLAB, we could find the following feasible solution to the LMIs (36):
Hence, system (41) has a unique equilibrium point and the unique equilibrium point is globally MittagLeffler stable by Corollary 3. In other words, system (39) and system (40) are globally asymptotically projective synchronized. In numerical simulation, the initial values for the drive system (39) are selected as \(x_{10}=1.5+0.5 \imath +2.5\jmath +3\kappa \) and \(x_{20}=1.50.5\imath 2.5\jmath 3 \kappa \), and the initial values for the drive system (40) are selected as \(y_{10}=2.53\imath 3\jmath 2\kappa \) and \(y_{20}=2.5+3\imath +3\jmath +2\kappa \).
At first, we consider the global MittagLeffler complete projective synchronization between system (39) and (40). Namely we choose the projective coefficient matrix
Figures 3 and 4 show the time response of four parts of state of drive system (39) and response system (40) with control. The phase plot of drive–response system (39) and (40) with control is shown in Figs. 5. Moreover, Fig. 6 shows the time responses of four parts of state variable of the error system (41). In addition, Figs. 7 and 8 show the time response of four parts of state of drive system (39) and response system (40) without control. Figure 9 shows the phase plot of drive–response system (39) and (40) without control.
In the same way, we choose the projective coefficient matrix
Then system (39) and system (40) realize global MittagLeffler projective antisynchronization. Figures 10 and 11 show the time response of four parts of states of drive system (39) and response system (40) with the projective coefficient matrix (43) under control. Figure 12 shows the phase plot of drive–response system (39) and (40) with the projective coefficient matrix (43) under control. Moreover, Fig. 13 shows the time response of four parts of states variable of error system (41).
Lastly, we consider the global projection synchronization. Then we choose the projective coefficient matrix
Figures 14 and 15 show the time response of four parts of states of drive system (39) and response system (40) with the projective coefficient matrix (44) under control. Figure 16 shows the phase plot of drive–response system (39) and (40) with the projective coefficient matrix (44) under control. Figure 17 shows the time response of four parts of states variable of error system (41).
Remark 7
When we choose $\Upsilon =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and $K=\left(\begin{array}{cc}4& 0\\ 0& 4\end{array}\right)$, the controller is the same as the one in the literature [44]. By using the LMI condition (3) in this paper we could calculate that \(\lambda (N)= 16.0622\), −16.0622, −15.5803, −15.5803, −9.7406, −9.7406, −7.9506, −7.9506. We find system (39) and (40) will be complete synchronization. But when we use Theorem 6 in [44] to test the synchronization, we cannot find any positive constant ρ such that
In fact, for all \(\rho >0\), we have \(2(c_{1}+\sigma _{1})\sum_{q=1} ^{n}\rho ^{1}(a_{1q}^{R}+a_{1q}^{I}+a_{1q}^{J}+a_{1q}^{K}) \sum_{q=1}^{n}\rho (a_{q1}^{R}+a_{q1}^{I}+a_{q1}^{J}+a_{q1} ^{K})=109.28\rho ^{1}8.79\rho \leq 7.58\) and \(2(c_{2}+\sigma _{2}) \sum_{q=2}^{n}\rho ^{1}(a_{2q}^{R}+a_{2q}^{I}+a_{2q}^{J}+a_{2q} ^{K})\sum_{q=1}^{n}\rho (a_{q2}^{R}+a_{q2}^{I}+a_{q2}^{J}+a _{q2}^{K})=108.72\rho ^{1}9.21\rho \leq 7,44\). So Theorem 6 in [44] cannot check system (39) and (40) realizes complete synchronization.
Conclusions
In this paper, some dynamical behaviors, including global MittagLeffler stability, robust stability and projection synchronization, for FQVNNs are studied. A LMI condition is given for MittagLeffler stability of FQVNNs by Lyapunov stability theory and homeomorphic mapping theory. Based on this, a modulus inequality technique of quaternions is used to study the robust stability of FQVNNs, and obtain a sufficient LMI condition for robust stability of FQVNNs. Moreover, the LMI condition for global MittagLeffler projection synchronization between FQVNNs is also given by the application of the projective synchronization theory. In addition, two corollaries for MittagLeffler stability and projective synchronization are given to make the validity of the conditions can be tested by the mathematical software MATLAB. Finally, three examples are given to substantiate the effectiveness of the theoretical results.
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Correspondence to Xiaofeng Chen.
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Keywords
 Fractionalorder quaternionvalued neural networks
 Global MittagLeffler stability
 Projective synchronization
 Robust stability
 Linear matrix inequality