This note solves the quasi-stability/synchronization for memristive neural networks on time scales. A novel Lemma is given to estimate the modulus of the quaternions, which avoids comparing the "size" of quaternions. Subsequently, a new scale-limited Halanay inequality is given to solve the quasi-synchronization control of the system in the time-scale domain. The validity of the theoretical results is illustrated by two examples.

This paper is concerned with global polynomial state estimation issue of memristive neural networks which take quaternion-valued parameters and fuzzy terms into account. First, via drawing support from quaternion-valued norm, an easily analyzable estimation error model is established, which overcomes the complexity brought by the system parameters. Then, a simple feedback controller is designed aiming to obtain the polynomial stability conditions. It is worth noting that, several algebraic forms of polynomial stability criteria for the error system proposed are achieved by applying the quaternion-valued norm, each of these criteria is represented by algebraic inequality, which facilitates validation. Ultimately, illustrative examples are given to show the effectiveness of the theoretical results.

In the dynamic analysis of memristive neural networks, most of existing conclusions on stability and synchronization cannot take into account both discrete and continuous cases simultaneously. However, some existing results of introducing time-scales theory can only provide Lagrange synchronization results. In this paper, the Lagrange exponential stability and the synchronization of memristive neural networks (MNNs) on time scales are considered. Firstly, the Lagrange exponential stability theory of MNNs is established by means of two-modal analysis. Secondly, a new scale-limited Halanay inequality lemma is constructed to research the synchronization control of the driving-response MNNs in the time-scale domain. Finally, a unified time-scale dynamic analysis tool set is developed, which can deal with the stability and synchronization problems under mixed time scales. Some examples are presented to demonstrate the effectiveness of the theorems, which are subsequently applied to the encryption and decryption of images.

. This paper illustrates the dissipative and dissipativity analysis for the fractional-order memristive system with quaternion terms. The purpose is to derive some conditions that are capable of guaranteeing the stability and dissipativity of the system. By resorting to a novel functional, sufficient conditions for the solvability of the above problem are established in the form of algebraic inequality and linear matrix inequality (LMI). In addition, based on the linear fractional difference system, the dissipativity conclusion is obtained, in which, the globally attractive sets is figured out as well. Finally, three examples are given to show the application of our proposed methods.

This study proposes a non‐decomposition method to examine the anti‐synchronization control of quaternion‐valued memristive neural networks with fuzzy terms. Using nonlinear scalarization method, the entire analysis does not use reduced order conversion, nor does it involve the separation of real and imaginary parts, but directly focuses on the original system, which preserved the integrity of the quaternion‐valued system. Furthermore, by means of the definition of , the quaternion‐valued memristive system is translated into a robust system with uncertain terms, which advanced some existing conclusions. Subsequently, sufficient conditions are derived to ensure the error system is exponential stable. Finally, examples are presented to demonstrate the proposed results.

This paper investigates the quasi-stability/synchronization problem for quaternion-valued fuzzy memristive system subject to fuzzy terms. Via a new exponentially quasi-stability definition, one can estimate the convergence rate more effectively, besides, the system can adjust the converge rate by modifying the controller. Then, with aim to utilize a new lemma generalized from Halanay inequality, some improved results are presented to achieve the quasi-stability/synchronization conclusions, which not only avoid considering the "magnitude" of quaternion, but also gives a more flexible criteria. Finally, example is given to substantiate the feasibility of the developed scheme.

The problems of exponential quasi-stabilization of quaternion-valued memristive networks are further studied in this article. Different from the common Lyapunov function method in the literature, where the conclusions are derived based on the algebraic inequality, in this article, a novel norm (Frobenius norm) is constructed by using the Lyapunov function, which avoid breaking the integrity of the system. In addition, the Frobenius norm are applied to derive new quasi-stabilization conditions. Less conservative results are obtained since the Frobenius norm contains more system information. Examples demonstrate the correctness of the developed new results.

This work explores the dissipativity and quasi-synchronization of a fuzzy memristive model with quaternion parameters and fractional derivatives. First, a fuzzy memristive model is presented through fuzzy logic. Second, by constructing innovative Lyapunov functional, the dissipativity and quasi-synchronization criteria are obtained in one-norm forms. As one knows, it is arduous to determine which is bigger between two quaternions, while in this paper, this topic is avoided by introducing the fuzzy terms, thus, in contrast to some existing results, this kind of treatment method is much easier to manipulate. Eventually, the corresponding numerical verification is carried out.

This article addresses the issue of Lagrange stability of quaternion-valued memristive neural networks. First, by scale-limited Halanay inequality, new Lagrange stability algebraic conditions are obtained. Considering that the memristive connections are switching between two different parameters, thus, the memristive model is equivalent to a robust system by introducing some new matrices, we then use the generalized matrix measure to solve the time scales matrix norm issues, and all the derived scale-limited sufficient criteria not only apply to continuous-time system and their discrete-time analogs, but also satisfy the system with uncertain terms. Eventually, illustrations are addressed to reflect the solvability and practicability of the strategy.

The exponential stability and the synchronization of QMNN are considered in this article. First, a new exponentially stable definition is proposed, it effectively estimates the rate of convergence, and the neural networks can converge to the equilibrium point at any rate by modifying the controller. Then, a simple controller is proposed to achieve the synchronization goal by a new lemma generalized from Halanay inequality, and a set of improved conditions are presented to achieve the synchronization control. The theorems are verified by some simulations in the end. The conclusions obtained in this paper not only avoid considering the "magnitude" of quaternion but also give a more flexible criteria.

In this paper, the problem of passivity and dissipativity analysis are investigated for a class of fractional-order quaternion-valued fuzzy memristive neural networks. By constructing proper Lyapunov functional and employing inequality technique, several improved passivity criteria and dissipativity conclusions are established, which can be checked efficiently by use of some standard mathematical calculations. Different from previous results, involving the quaternions connections, our derivation avoid considering the "magnitude" of quaternion. Finally, two simulation examples based on the fuzzy model are given to demonstrate the effectiveness of the proposed techniques.

This article focuses on the input-to-state stability (ISS) issue of quaternion-valued memristive networks. Employing the quaternion norm tool and the Lyapunov method, two improved conclusions are developed for the continuous networks. After that, via the semidiscretization technique, a new discrete model is designed, and its ISS performance is discussed and subsequently recur to a nonlinear scalarization approach. Less conservative results are obtained since the nonlinear scalarization approach makes the quaternion interval meaningful. Simulations are presented to verify the validity of the outcomes.

In this article, we consider exponential estimation and passivity of memristive neural networks with quaternion parameters. A Takagi–Sugeno type rule is introduced into the quaternion memristive neural networks, which makes the system much easier. To achieve the exponential stability of the estimation error system, a proper controller is designed, which derived in the two norm form. Further, the quasi-state estimation condition is also considered. Along with Lyapunov theory, some criteria are obtained to achieve the exponential passivity of the discussed system.

In this paper, we present new conditions ensuring existence, uniqueness, and stabilization of the equilibrium point (EP) for a large class of quaternion-valued memristive neural networks. The results are applicable to both complex-valued memristive neural networks and quaternion-valued system with uncertain terms. Subsequently, by employing suitable controller, the synchronization control conclusion is also considered. It is worth mentioning that, the nonlinear scalarization approach is employed to ensure the closed convex hull consisted by the quaternion is meaningful, which preserves the integrity of the quaternion system and makes the conclusion more reasonable. Finally, this paper provides two examples, so that the superiority of the main results can be illustrated.

This article is devoted to solving the exponential synchronization problem of a new type of fuzzy memristive neural network with reaction-diffusion terms. By introducing adaptive laws, two controllers are designed. After combining the inequality technique with the Lyapunov function approach, some easily verified sufficient conditions are established to ensure the exponential synchronization of the reaction-diffusion fuzzy memristive system under the proposed adaptive scheme. In addition, by using the Hardy-Poincarè inequality, the diffusion terms are estimated associated with the information of the reaction-diffusion coefficients and the regional feature, which improves some existing conclusions. Finally, an illustrative example is presented to demonstrate the validity of the theoretical results.

This paper examines the quasi-stabilization and synchronization of memristive neural networks in high dimensional forms. First, new criteria for the existence of the equilibrium point (EP) are delivered by contraction mapping principle. Then, a new Lyapunov function described with one-norm form is imported for quasi-stabilization/synchronization of the system. Finally, two numerical examples containing simulations are given to demonstrate the effectiveness.

This work explores the exponentially quasi-synchronization control of quaternion-valued memristive neural networks. The entire analysis does not use reduced-order conversion, nor does it involve the separation of real and imaginary parts, but directly focuses on the original system, which preserved the integrity of the quaternion-valued system. First, the definition of √(x) , x∈ℚ is first introduce, which provides new methods to investigate quaternion-valued systems in a compact form. Second, using Frobenius norm, the derived conclusions contain much more information of the matrix elements compared with the traditional 1-norm or 2-norm. Eventually, numerical examples are carried out to verify the derived schemes.

This paper deals with the dissipative control for a class of quaternion-valued fuzzy memristive neural networks. By constructing proper Lyapunov functionals and using adaptive controller, the strictly (Q,S,R)-dissipative are characterized parametrically. Then, based on the algebraic inequality and linear matrix inequality (LMI) approach, sufficient conditions for the existence of the dissipative controllers are obtained. In addition, the nonlinear scalarization approach is developed, which can be employed to compare the “size” of two different quaternions, in this way, the convex closure proposed by the quaternion weights are meaningful. Finally, simulation examples are given to show the efficiency of the proposed methods.

This paper introduces a novel approach to study the quaternion‐valued memristive neural networks and uses this method to investigate the Lagrange stability of the memristive system. By utilizing the logical switching signal, a stop and go strategy is proposed, which makes the model translated into a system contain two different switching parameters; thus, the derived conclusions contain much more information of the states. Moreover, the difficulty of comparing the “size” of quaternions can be avoided. In the case of discrete‐time system, a very useful lemma is presented, which makes the analysis of the discrete‐time quaternion‐valued memristive network easy to operate. Through simulations, the main results regarding to the Lagrange stability are illustrated.