Synchronization and stability of delayed fractional-order memristive quaternion-valued neural networks with parameter uncertainties

Unified Analysis on the Global Dissipativity and Stability of Fractional-Order Multidimension-Valued Memristive Neural Networks With Time Delay

The unified criteria are analyzed on the global dissipativity and stability for the delayed fractional-order systems of multidimension-valued memristive neural networks (FSMVMNNs) in this article. First, based on the comprehensive knowledge about multidimensional algebra, fractional derivatives, and nonsmooth analysis, we establish the unified model for the studied FSMVMNNs in order to propose a more uniform method to analyze the dynamic behaviors of multidimensional neural networks. Then, by mainly applying the Lyapunov method, employing several new lemmas, and solving some mathematical difficulties, without any separation, we acquire the unified and concise criteria. The derived criteria have many advantages in a smaller calculation, lower conservatism, more diversity, and higher flexibility. Finally, we provide two numerical examples to express the availability and improvements of the theoretical results.

Quantized intermittent control tactics for exponential synchronization of quaternion-valued memristive delayed neural networks

This article studies the global exponential synchronization (GES) of quaternion-valued memristive delayed neural networks (QVMDNNs) by quantized intermittent control (QIC). Without decomposing the original systems into usual real-valued or complex-valued ones, the discussed system is directly processed in both cases of the differential inclusion theory. Based on two QIC strategies and applying Lyapunov functional method and inequality techniques, several new delay-dependent criteria in the form of real-valued algebraic inequalities are directly derived to assure the GES of the concerned system. Compared to the traditional feedback control, the QIC strategy can reduce the control expenses and shorten time to achieve the GES. Ultimately, two examples are given to validate the effectiveness and advantages of the proposed outcomes.

Artificial neural networks: a practical review of applications involving fractional calculus

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

Novel Inequalities to Global Mittag-Leffler Synchronization and Stability Analysis of Fractional-Order Quaternion-Valued Neural Networks

This article is concerned with the problem of the global Mittag-Leffler synchronization and stability for fractional-order quaternion-valued neural networks (FOQVNNs). The systems of FOQVNNs, which contain either general activation functions or linear threshold ones, are successfully established. Meanwhile, two distinct methods, such as separation and nonseparation, have been employed to solve the transformation of the studied systems of FOQVNNs, which dissatisfy the commutativity of quaternion multiplication. Moreover, two novel inequalities are deduced based on the general parameters. Compared with the existing inequalities, the new inequalities have their unique superiorities because they can make full use of the additional parameters. Due to the Lyapunov theory, two novel Lyapunov-Krasovskii functionals (LKFs) can be easily constructed. The novelty of LKFs comes from a wider range of parameters, which can be involved in the construction of LKFs. Furthermore, mainly based on the new inequalities and LKFs, more multiple and more flexible criteria are efficiently obtained for the discussed problem. Finally, four numerical examples are given to demonstrate the related effectiveness and availability of the derived criteria.

Improved approach to the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks based on new inequalities

This paper studies the problem of the global Mittag-Leffler synchronization for fractional-order multidimension-valued BAM neural networks (FOMVBAMNNs) with general activation functions (AFs). First, the unified model is established for the researched systems of FOMVBAMNNs which can be turned into the corresponding multidimension-valued systems as long as the state variables, the connection weights and the AFs of the neural networks are valued to be real, complex, or quaternion. Then, without any decomposition, the criteria in unified form are derived by constructing the new Lyapunov-Krasovskii functionals (LKFs) in vector form, combining two new inequalities and considering the easy controllers. It is worth mentioning that the obtained criteria have many advantages in higher flexibility, more diversity, smaller computation, and lower conservatism. Finally, a simulation example is provided to illustrate the availability and improvements of the acquired results.

Stochastic Memristive Quaternion-Valued Neural Networks with Time Delays: An Analysis on Mean Square Exponential Input-to-State Stability

In this paper, we study the mean-square exponential input-to-state stability (exp-ISS) problem for a new class of neural network (NN) models, i.e., continuous-time stochastic memristive quaternion-valued neural networks (SMQVNNs) with time delays. Firstly, in order to overcome the difficulties posed by non-commutative quaternion multiplication, we decompose the original SMQVNNs into four real-valued models. Secondly, by constructing suitable Lyapunov functional and applying It o ^ ’s formula, Dynkin’s formula as well as inequity techniques, we prove that the considered system model is mean-square exp-ISS. In comparison with the conventional research on stability, we derive a new mean-square exp-ISS criterion for SMQVNNs. The results obtained in this paper are the general case of previously known results in complex and real fields. Finally, a numerical example has been provided to show the effectiveness of the obtained theoretical results.

Global Stability Analysis of Fractional-Order Quaternion-Valued Bidirectional Associative Memory Neural Networks

We study the global asymptotic stability problem with respect to the fractional-order quaternion-valued bidirectional associative memory neural network (FQVBAMNN) models in this paper. Whether the real and imaginary parts of quaternion-valued activation functions are expressed implicitly or explicitly, they are considered to meet the global Lipschitz condition in the quaternion field. New sufficient conditions are derived by applying the principle of homeomorphism, Lyapunov fractional-order method and linear matrix inequality (LMI) approach for the two cases of activation functions. The results confirm the existence, uniqueness and global asymptotic stability of the system’s equilibrium point. Finally, two numerical examples with their simulation results are provided to show the effectiveness of the obtained results.

Global Mittag–Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks

This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued systems is divided into four fractional-order real-valued systems. By using the method of Lyapunov fractional-order derivative, fractional-order differential inclusions, set-valued maps, several global Mittag–Leffler stability and stabilization conditions of considered FOQVMNNs are established. Two numerical examples are provided to illustrate the usefulness of our analytical results.