Stability Analysis of Quaternion-Valued Neutral-Type Neural Networks with Time-Varying Delay

This paper addresses the problem of global μ-stability for quaternion-valued neutral-type neural networks (QVNTNNs) with time-varying delays. First, QVNTNNs are transformed into two complex-valued systems by using a transformation to reduce the complexity of the computation generated by the non-commutativity of quaternion multiplication. A new convex inequality in a complex field is introduced. In what follows, the condition for the existence and uniqueness of the equilibrium point is primarily obtained by the homeomorphism theory. Next, the global stability conditions of the complex-valued systems are provided by constructing a novel Lyapunov–Krasovskii functional, using an integral inequality technique, and reciprocal convex combination approach. The gained global μ-stability conditions can be divided into three different kinds of stability forms by varying the positive continuous function μ(t). Finally, three reliable examples and a simulation are given to display the effectiveness of the proposed methods.


Introduction
As is well known, with the rapid development of electronic information science, complex-valued signals appear frequently in engineering practice.The application fields of complex-valued neural networks (CVNNs) are also becoming increasingly extensive: for instance, automatic control, eddy current defect detection, image processing, object recognition, frequency-domain blind source separation, and signal processing (see, e.g., [1][2][3][4][5][6]).Hence, many scholars are directing much attention to studying the dynamic behavior of CVNNs, and lots of important results have been reported in the literature.The exponential stability of complex-valued BAM neural networks was studied based on the differential inclusion theory and the properties of homeomorphism [7].The synchronization problem for CVNNs with time delays was discussed in [8,9].Following these results, in [10,11], the problem of extended dissipative synchronization of CVNNs was also discussed.In [12], the Lagrange stability of CVNNs was studied by using a transformation in which the CVNN is rewritten as a first-order differential system.In [13][14][15][16], the authors studied the impact of impulses on the stability of CVNNs with time-varying delays, and they obtained ample conditions for the CVNNs to ensure exponential convergence.Moreover, fractional complex-valued neural networks (FCVNNs) have certain advantages when describing dynamical properties.In [17], Huang studied local asymptotical stability and Hopf bifurcation, and the condition for the emergence of bifurcation was obtained.
In fact, the quaternion as an extension of a complex-valued system can also be applied to engineering practices.This issue has aroused the interest of scholars.After an active exploration, scholars found that the quaternion can also play a very important role in engineering, mainly on the basis of its advantages in rotation and direction modeling.For example, a data covariance model using a quaternion form was proposed to estimate its wavenumber and polarization parameters, similar to a music algorithm [18].In addition, quaternions are used to define Fourier transforms that are suitable for color images.It was also shown that the transformation can be calculated using two standard complex fast Fourier transforms [19].
In recent years, it has become gradually more common to discuss the quaternion-valued neural network (QVNN) as an extension of the CVNN because of the following facts.On the basis of Liouville's theorem [20], each bounded function must be constant, i.e., the activation function of CVNNs cannot have boundaries and be analytic at the same time unless it is a constant.However, the activation function of QVNNs can be bounded and analytic at the same time, as applied in [21], but how to choose the activation function of QVNN is a difficult problem.The analyticity of general quaternion-valued functions has not been rigorously examined in the quaternion field.To ensure that the class of quaternion-valued functions is analyzed, strict Cauchy-Riemann-Fueter (CRF) and generalized Cauchy-Riemann (GCR) conditions only pledge that the global analysis of quaternion-valued functions is a linear function and a constant, respectively.To overcome this difficulty, References [22,23] give some very important conditions for a partial change to the Cauchy-Riemann-Fueter condition and the local analysis condition-namely, the local analyticity condition (LAC)-to ensure that the quaternion-valued functions are bounded and analytic at the same time.This technique, which provides more flexibility in choosing the activation function of QVNNs, is significant progress.Until now, quaternion algebra has been successfully applied to communications problems and signal processing, such as color image processing [24] and wind forecasting [25].Since then, numerous scholars have produced many excellent results in the field of QVNNs (see, e.g., [26][27][28][29] and literature referenced therein).QVNN was changed into two complex-valued systems by using a simple transformation, and [26] reduced the complexity of computation generated by the non-commutativity of quaternion multiplication.With homeomorphism theory, Reference [27] proved the existence of the equilibrium point of QVNNs and provided ample conditions for global robust stability.In [28], the pseudo-major period synchronization problem of quaternion-valued cellular neural networks (QVCNNs) was also studied.The existence of pseudo almost periodic functions was proved, and the global exponential synchronization of QVCNNs was obtained by designing the controller and combining Lyapunov functions.
On the other hand, the neutral-type systems not only consider the past state but also specifically involve the influence of changes in past states on the current state.Due to this feature, neutral-type systems have become the subject of extensive research by many scholars (see [30][31][32][33][34][35][36][37][38]).Furthermore, neutral systems have many applications in practical engineering, including heat exchangers, population ecology, and so on (see [39,40]).Many neural networks can be regarded as special cases of neutral neural networks, and most of the neural networks can be transformed into neutral neural networks for research (see [41][42][43]).It can be seen that the neutral neural network has great research value and potential significance.Nevertheless, to the best of the authors' knowledge, for QVNNs with time-varying delays, there is no research in the literature for the global µ-stability of quaternion-valued neutral-type neural networks (QVNTNNs) at this time.
All of the above factors motivate our research.This article is intended to discuss the µ-stability of QVNTNNs.The remainder is divided into the following sections to elaborate.In the second part, the fundamental definition of quaternion is given.In the third part, we first introduce the QVNTNN model.Then, some important definitions and lemmas are provided, and the new extended convex inequality is obtained for the first time in this paper.In the fourth part, using the homeomorphism theory, we firstly obtain a new condition for the existence and uniqueness of the equilibrium point, and the global µ-stability criterion for QVNTNNs is provided using the Lyapunov functional theory combined with some inequality techniques.Based on the obtained stability results, power-stability, log-stability, and exponential stability are given as corollaries.In the fifth part, the effectiveness and feasibility of the method in this paper are illustrated by three examples.In the sixth part, we draw conclusions of the article.
Notations: Some significant symbols used throughout this paper are considerably standard.R n×m denotes the collection of all n × m real-valued matrices.C n×m denotes the collection of all n × m complex-valued matrices.Q n×m denotes the collection of all n × m quaternion-valued matrices.diag(• • • ) denotes a block-diagonal matrix.• denotes the Euclidean vector norm.SC n (Q) denotes the collection of all quaternion positive matrices and quaternion self-conjugate matrices.p denotes a quaternion-valued function, and p denotes the conjugate of p.The superscript * denotes the transpose of a matrix or a vector.For any matrix G, λ max (G)(λ min (G)) denotes the largest (smallest) eigenvalue of G.

Definition of Quaternion
The quaternion consists of four parts, one of which is a real number and three of which are imaginary numbers, (i, j, and k).Generally, the quaternion is defined by a vector q, where q belongs to the four-dimensional vector space.We use the following form to represent the quaternion q = q 0 + q 1 i + q 2 j + q 3 k, where q v (v = 0, 1, 2, 3) are real numbers and i, j, k satisfy the multiplication table formed by The above representations are said to be the Hamilton rule.We see that the multiplication of the quaternion is not interchangeable.Similar to the definition of complex, q is defined as the conjugate of the quaternion q ∈ Q n .q = q 0 − q 1 i − q 2 j − q 3 k, For any q ∈ Q n , |q| = √ q q = q 2 0 + q 2 1 + q 2 2 + q 2 3 .q can be expressed as q = c 1 + c 2 j with each q ∈ Q n , where c 1 , c 2 ∈ C n .

Problem Statement and Preliminaries
Firstly, the delayed QVNTNN is introduced by the following where y(t) = (y 1 (t), y 2 (t), . . ., y n (t)) * ∈ Q n is the state vector, and p( ) = (p( ), . . ., p n ( ) where Particularly, jT = T j or jT j = T for any complex matrix T ∈ C n×n .Assumption 2. The neuron activation function p v (•) and pv (y(•)) (v = 1, 2) satisfy the Lipschitz condition for any y, y ∈ C n , y = y .There exist constants L Assumption 3.According to the stability of the theorem in [44] for neutral systems, we assume that the radius of C is smaller than 1.
Remark 1.The gained µ-stable conditions can be transformed as power-stability, log-stability, and exponential stability by varying the positive continuous function µ(t).

Definition 2 ([45]
).For a function e t , which is positive and continuous, let t → +∞; it is clear that e t → +∞.Then, there exists a positive constant ϕ for all t > 0 such that the following inequality holds: and the QVNTNN (Equation ( 1)) is called exponentially stable.

Definition 3 ([45]
).For a function t , which is positive and continuous, let t → +∞; it is clear that t → +∞ if there exists a constant ϕ > 0 such that the following inequality holds: and the QVNTNN (Equation (1)) is power-stable.
Lemma 1 ([46]).For given a Hermitian matrix W > 0, the following inequality holds for all continuously differentiable functions φ in [f, g] → C n×n : where Lemma 2 ([26]).If each given matrix G ∈ SC n (Q), then each eigenvalue of matrix G is real.Lemma 3 ([47]).If there exists a continuous mapping M(y): C n → C n and it satisfies the following conditions (1) M(y): and vectors ξ i which satisfy then, the following inequality holds: Proof.For i = 2, it is easy to see that the following inequality The situation of i = n can also be established with a similar method.Here, the proof processing is omitted.Remark 2. Clearly, Lemma 4 is an extension of Lemma 2 in [48], which just considers the application in the real number field.Lemma 4 can be applied to the complex field.Therefore, the range of application of Lemma 4 is wider than that given in [48].This paper is further extended by the literature [48] so that it can be applied to the complex number field.Thus, one can conclude that the range of application for Lemma 4 is wider and more practical.

Main Results
In the following content, we first present the condition for the existence and uniqueness of the equilibrium point for the system in Equation ( 1).
Theorem 1.On the basis of Assumptions 1 and 2, the system in Equation ( 1) has a unique equilibrium point if there exists a positive diagonal matrix V i (i = 1, 2, . . ., 6) and the following LMIs are satisfied where Proof.According to Assumption 1, Equation ( 1) can be rewritten in the following form ( To prove the existence and uniqueness of the solution, we need to construct a mapping which combines the information of the system in Equation ( 3), and it can be written as follows: where If y is an equilibrium point of the system in Equation ( 1), in light of Assumptions 1 and 3, let y = y1 + y2 j; then, y satisfies the following equation In light of Lemma 4, if M(y) satisfies the homeomorphic mapping on C n , then we can find conditions to guarantee that there exists a unique equilibrium point for the system in Equation ( 1).
Next, the proof is divided into two sections.
In the first place, we need to prove that M(y 1 , y 2 ) is an injective.If we choose two points, (y 1 , y 2 ) * , (y 1 , y 2 ) * ∈ C n and (y 1 , y 2 ) = (y 1 , y 2 ), in light of the definition of the activation function given by Assumption 2, we have p(y 1 , y 2 ) = p(y 1 , y 2 ).
In the second place, we need to prove that M(y 1 , y 2 ) → ∞ as (y 1 , y 2 ) → ∞.Let (y 1 , y 2 ) = (0, 0); then, we have From the Cauchy-Schwarz inequality, we have while (y 1 , y 2 ) = 0. So, we have Therefore, we clearly know that M(y 1 , y 2 ) → ∞ as (y 1 , y 2 ) → ∞.Thus, the conditions of Lemma 3 are satisfied, and M(y 1 , y 2 ) is a homeomorphism mapping.Hence, from Corollary 1 in [49], the condition for the existence of a unique equilibrium point of the system in Equation ( 1) is proved.
In the following content, we present the conditions for global µ-stability criteria for the system in Equation (1).Firstly, suppose that y is the unique equilibrium point of the QVNTNN (Equation ( 1)), where y = ( y, y, . . ., y) * .According to Assumption 1 and the transformation ỹ = y − y, the system in Equation ( 1) can be rewritten as the following: For the sake of convenience, in this paper, some symbols are defined as follows: u)du, Now, we present our main results in the following theorem.
Theorem 2. Assume that Assumptions 1 and 2 hold.For a given positive constant ν, the equilibrium point of QVNTNNs (Equation ( 1)) is µ-stable if there exist positive definite Hermitian matrices such that the following LMIs hold where Proof.Let us choose a new Lyapunov-Krasovskii functional for the system in Equation ( 13) as follows: where Applying Lemma 1 to the integral term in Equation ( 16) yields , where Furthermore, due to Si ≥ 0(i = 1, 2), according to Lemma 4, we can easily get .
On the other hand, for any diagonal matrices N i ≥ 0(i = 1, 2, 3, 4), it follows from Assumption 2 that with ).The following zero inequalities are introduced with appropriate dimensional complex-valued matrices N 5 ≥ 0 and N 6 ≥ 0: Combining ∑ 4 i=1 Vi with Equations ( 17) and ( 18), we can easily get that where Φ, Ω, S1 , and S2 are defined in Theorem 2. Consequently, according to Equation ( 14), we have Combined with Lemma 2, we claim that Λ min (P ) is constant.Then, from Equation ( 15), one can get for 0 ≤ t 0 ≤ t, and we have where ℘ = V(0) Λ min (P ) .By the above derivation, it is obvious that Definition 1 is satisfied, and the origin point of QVNTNNs (Equation ( 1)) is µ-stable.
Corollary 1. Assume that Assumptions 1 and 2 hold.Given a positive constant ν, the equilibrium point of QVNTNNs (Equation ( 1)) is globally exponentially stable if there exist positive definite Hermitian matrices 6), and if µ(t) is a nonnegative function which belongs to L 2 [0, ∞) such that Φ, Ω, and Si (i = 1, 2) in Theorem 2 hold, where Proof.Taking µ(t) = e t , we can obtain μ(t) On the basis of the above discussion, it is clear that the results are derived directly via Theorem 2. This proof is immediately completed.Corollary 2. Assume that Assumptions 1 and 2 hold.Given a positive constant ν, the equilibrium point of QVNTNNs (Equation ( 1)) is globally power-stable if there exist positive definite Hermitian matrices 6), and if µ(t) is a nonnegative function which belongs to L 2 [0, ∞) such that Φ, Ω, and Si (i = 1, 2) in Theorem 2 hold, where Proof.Taking µ(t) = t , for any t ≥ 2 max{1, ν}, we can obtain μ(t) By the above computation, it is concluded that the conditions in Theorem 2 are still satisfied.The proof is completed.Corollary 3. Assume that Assumptions 1 and 2 hold.Given a positive constant ν, the equilibrium point of QVNTNNs (Equation ( 1)) is globally log-stable if there exist positive definite Hermitian matrices 6), and if µ(t) is a nonnegative function which belongs to L 2 [0, ∞) such that Φ, Ω, and Si (i = 1, 2) in Theorem 2 hold, where Proof.Taking µ(t) = ln( t + 1), for any t ≥ ( e−1 ) + ν, we have μ(t) From the above expressions, and based on the Theorem 2, one can conclude that the conditions in Corollary 3 can be easily achieved.Thus, this completes the proof.Remark 3. Compared with the existing literature (see, e.g., [27,28,30]), we use the reciprocal convex combination approach combined with the free-weighting matrix method for getting Theorem 2. In this way, the time-delay information is fully explored and can be a reduced conservative result.Remark 4. By Theorem 2, we obtain the stability criterion of global µ-stability, and then we can generalize the results to the global exponential stability, global power-stability, and global log-stability.
Remark 5. Since the delay-dependent stability conditions are always less conservative than the delay-independent stability conditions, this paper mainly considered the delay-dependent stability for the systems with bounded time-varying delays.In fact, the stability conditions of QVNTNNs that are unbounded time-varying can also established with a similar method.Moreover, the stability conditions in this paper are also suitable for unbounded time-varying delays depending on the properties of the QVNTNN itself.

Numerical Example
In order to show the effectiveness and advantages of the proposed method, three interesting numerical examples are given as follows.

part of y 1 4th part of y 2 Figure 3 .
Figure 3.The four parts of the state trajectories for the QVNTNNs (Equation (1)) in Example 3.