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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.

In practical applications, stochastic effects are normally viewed as the major sources that lead to the system’s unwilling behaviours when modelling real neural systems. As such, the research on network models with stochastic effects is significant. In view of this, in this paper, we analyse the issue of robust stability for a class of uncertain complex-valued stochastic neural networks (UCVSNNs) with time-varying delays. Based on the real-imaginary separate-type activation function, the original UCVSNN model is analysed using an equivalent representation consisting of two real-valued neural networks. By constructing the proper Lyapunov–Krasovskii functional and applying Jensen’s inequality, a number of sufficient conditions can be derived by utilizing It o ^ ’s formula, the homeomorphism principle, the linear matrix inequality, and other analytic techniques. As a result, new sufficient conditions to ensure robust, globally asymptotic stability in the mean square for the considered UCVSNN models are derived. Numerical simulations are presented to illustrate the merit of the obtained results.

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.

We study the robust dissipativity issue with respect to the Hopfield-type of complex-valued neural network (HTCVNN) models incorporated with time-varying delays and linear fractional uncertainties. To avoid the computational issues in the complex domain, we divide the original complex-valued system into two real-valued systems. We devise an appropriate Lyapunov-Krasovskii functional (LKF) equipped with general integral terms to facilitate the analysis. By exploiting the multiple integral inequality method, the sufficient conditions for the dissipativity of HTCVNN models are obtained via the linear matrix inequalities (LMIs). The MATLAB software package is used to solve the LMIs effectively. We devise a number of numerical models and their empirical results positively ascertain the obtained results.Mathematics 2020, 8, 595 2 of 22 models has been highly focused, resulting in many research studies with comprehensive results [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. On the other hand, it is important to investigate the stability of NN models with the effects of linear fractional uncertainties. Because, when practical systems are modelled, uncertainties of system parameters are often included. From the application point of view, it is important to investigate NN models with linear fractional uncertainties. Several methods for analyzing the dynamical properties of NN models with linear fractional uncertainties have recently been proposed [23,24,39]. By using the Lyapunov function, the robust stability of delayed NN models has been studied with linear fractional uncertainties [23]. In [39], several sufficient conditions have been derived. The study focuses on impulsive NN models, whereby the problem of state feedback synchronization control considering linear fractional uncertainties along with mixed delays has been tackled.An essential property pertaining to dynamical systems is the dissipativity theory. It provides more knowledge than stability. This is because stability analysis is normally strictly related to the phenomenon of energy dissipation or loss. Besides that, the dissipativity theory offers a critical methodology for designing control systems through an input-output representation using system energy-related contemplations. As a result, many publications on the dissipativity analysis of NN models are available in the literature [26][27][28][29][30][31][32][33][34][35][36][37]40]. As an example, a number of new conditions with respect to the (Q, S, R) dissipativity criteria, global exponential dissipativity, and global dissipativity have been developed for a class of CVNN models in [31,37]. In [32], the use of Dini derivative concepts has resulted in novel sufficient conditions for the dissipativity of complex-valued bi-directional associative memory NN models. The dissipativity of discrete-time systems has been studied in [34]. A new concept of dissipativity has been introduced to describe the changes in subsystems and dissipation of energy of the considered system. Most of the existing stu...