Search Results (23)

To address the challenge of effectively stabilizing inherently unstable hybrid stochastic McKean–Vlasov equations (HMVSDEs) while simultaneously minimizing control costs, this paper proposes a novel control strategy termed aperiodically intermittent control (AIC). Under the global Lipschitz condition, we first establish the existence and uniqueness theorem for the solutions to HMVSDEs. Subsequently, we derive a generalized It$\widehat{\mathrm{o}}$ formula for HMVSDEs, based on which we construct a Lyapunov functional that explicitly incorporates both the law (distribution) of the solution and the underlying Markovian switching process. By employing the Lyapunov functional method, we rigorously construct AIC for the unstable HMVSDEs and analyze the mean-square exponential stability of the controlled system. Furthermore, we demonstrate the applicability of the proposed AIC strategy through a mean-field stochastic Cohen–Grossberg–Hopfield neural network model. Finally, a numerical example is provided to illustrate the effectiveness and practical feasibility of the developed control approach. Full article

This paper addresses the stabilization problem for a class of fractional-order memristive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays under an intermittent boundary control framework. Two scenarios are considered: systems without parametric uncertainties, for which asymptotic stability is established, and systems with uncertainties, for which robust asymptotic stability is ensured. By constructing appropriate Lyapunov functionals and employing Wirtinger-type inequalities, the fractional Razumikhin approach, and key properties of the Mittag–Leffler function, sufficient stability conditions are derived in terms of linear matrix inequalities with reduced conservatism. Furthermore, the effects of time-varying delays and control activation intervals on the stabilization performance are systematically investigated. The effectiveness and advantages of the proposed control methodology are validated through numerical simulations. Full article

The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons are modeled by generalized Caputo fractional derivatives with respect to another function (GCFDF). We consider a time-dependent delay and a switching rule in the model, which specifies when to switch the system at the initially given times. The switching rule is a piecewise constant function, and its points of discontinuity are the lower limits of the applied GCFDF on the corresponding intervals. We develop theoretical tools for GCFDF, starting with an important inequality for estimating that derivative on quadratic functions. We define the global Mittag–Leffler synchronization and obtain sufficient conditions based on the Lyapunov method, using a Razumikhin condition and quadratic functions. Full article

This paper investigates the existence and global exponential stability of almost periodic solutions in a class of impulsive Cohen–Grossberg-type bidirectional associative memory (BAM) neural networks with time-varying delays. Real neural systems often experience sudden perturbations and nonuniform temporal interactions, leading to complex oscillatory behaviors. To capture these effects, a new impulsive Cohen–Grossberg BAM model is developed that integrates both delays and impulsive influences within a unified framework. Using the theory of almost periodic functions, fixed point methods, and impulsive differential inequalities, new sufficient conditions are derived for the existence and stability of almost periodic solutions. A Lyapunov functional combined with a generalized Gronwall-type inequality provides rigorous global exponential stability criteria. Numerical simulations confirm the theoretical analysis. The results extend existing studies and offer new insights into how delay and impulsive factors jointly shape the stability and dynamics of hybrid neural systems, contributing to the design of robust and delay-tolerant neural architectures. Full article

This paper establishes a rigorous theoretical framework for analyzing the existence and uniqueness of solutions to Cohen–Grossberg bidirectional associative memory neural networks (CGBAMNNs) incorporating four distinct types of time-varying delays: leakage, neutral, distributed, and transmission delays. This study makes three key contributions to the field: First, it overcomes the fundamental challenge posed by the system’s inherent inability to be expressed in vector–matrix form, which previously limited the application of standard analytical techniques. Second, the work develops a novel and generalizable methodology that not only proves sufficient conditions for solution existence and uniqueness but also, for the first time in the literature, provides an explicit representation of the unique solution. Third, the proposed framework demonstrates remarkable extensibility, requiring only minor modifications to be applicable to a wide range of delayed system models. Theoretical findings are conclusively validated through numerical simulations, confirming both the robustness of the proposed approach and its practical relevance for complex neural network analysis. Full article

There are three principal novelties in the present investigation. It is the first time Cohen–Grossberg-type neural networks are considered with the most general delay and advanced piecewise constant arguments. The model is alpha unpredictable in the sense of electrical inputs and is researched under the conditions of alpha unpredictable and Poisson stable outputs. Thus, the phenomenon of ultra Poincaré chaos, which can be indicated through the analysis of a single motion, is now confirmed for a most sophisticated neural network. Moreover, finally, the approach of pseudo-quasilinear reduction, in its most effective form is now expanded for strong nonlinearities with time switching. The complexity of the discussed model makes it universal and useful for various specific cases. Appropriate examples with simulations that support the theoretical results are provided. Full article

This article focuses on the problem of finite-time and fixed-time synchronization for Cohen–Grossberg neural networks (CGNNs) with time-varying delays and memristor connection weights. First, through a nonlinear transformation, an alternative system is derived from the Cohen–Grossberg memristor-based neural networks (MCGNNs) considered. Then, under the framework of the Filippov solution and by adjusting a key control parameter, some novel and effective criteria are obtained to ensure finite-time or fixed-time synchronization of the alternative networks via the unified control framework and under the same conditions. Furthermore, the two types of synchronization criteria are derived from the considered MCGNNs. Finally, some numerical simulations are presented to test the validity of these theoretical conclusions. Full article

In this paper, we focus on h-manifolds related to impulsive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays. By constructing a new Lyapunov-type function and a comparison principle, sufficient conditions that guarantee the global practical exponential stability of specific states are established. The states of interest are determined by the so-called h-manifolds, i.e., manifolds defined by a specific function h, which is essential for various applied problems in imposing constraints on their dynamics. The established criteria are less restrictive for the variable domain and diffusion coefficients. The effect of some uncertain parameters on the stability behavior is also considered and a robust practical stability analysis is proposed. In addition, the obtained h-manifolds’ practical stability results are applied to a bidirectional associative memory (BAM) neural network model with impulsive perturbations and time-varying delays. Appropriate examples are discussed. Full article

The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons is modeled by a Riemann–Liouville fractional derivative with respect to another function and an appropriate initial condition is set up. Some inequalities about both the quadratic function and the absolute values functions and their fractional derivatives with respect to another function are proved and they are based on an appropriate modification of the Razumikhin method. These inequalities are applied to obtain the bounds of the norms of any solution of the model. In particular, we apply the squared norm and the absolute values norms. These bounds depend significantly on the function applied in the fractional derivative. We study the asymptotic behavior of the solutions of the model. In the case when the function applied in the fractional derivative is increasing without any bound, the norms of the solution of the model approach zero. In the case when the applied function in the fractional derivative is equal to the current time, the studied problem reduces to the model with the classical Riemann–Liouville fractional derivative and the obtained results gives us sufficient conditions for asymptotic behavior of the solutions for the corresponding model. In the case when the function applied in the fractional derivative is bounded, we obtain a finite bound for the solutions of the model. This bound depends on the initial function and the solution does not approach zero. An example is given illustrating the theoretical results. Full article

This work is dedicated to exploring the globally exponential stability of anti-periodic solutions in inertial CGNNs that incorporate time delays. This is based on a strategic variable substitution to transform the complex system into a first-order differential equation. By leveraging the Lyapunov functional and demonstrating uniformly converging properties, we establish sufficient conditions that guarantee the existence and global exponential stability of anti-periodic solutions for the system. Finally, examples are presented to illustrate the effectiveness of the obtained theoretical results. This work contributes significantly to enhancing our understanding of the stability dynamics in neural networks with time delays and provides valuable insights for applications across various fields. Full article

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria. Full article

Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values were obtained. These Lyapunov functions were combined with a modification of the Razumikhin method to study the stability properties of the Cohen–Grossberg model of neural networks with both time-variable and continuously distributed delays, time-varying coefficients, and RLGFDs. The initial-value problem was set and studied. Upper bounds by exponential functions of the solutions were obtained on intervals excluding the initial time. The asymptotic behavior of the solutions of the model was studied. Some of the obtained theoretical results were applied to a particular example. Full article

As an essential dynamic behavior, the synchronization of inertial Cohen–Grossberg neural networks (ICGNNs) has received considerable attention due to its successful applications in neural cryptography, public channel cryptography, security communications, and image encryption. In this article, the $\alpha$-synchronization of a class of non-autonomous unbounded delayed inertial Cohen–Gossberg neural networks with delayed impulses is investigated. Firstly, several non-autonomous impulsive differential inequalities are established, where unbounded delays, delayed impulses, and time-variable coefficients are incorporated. Subsequently, based on the proposed impulsive differential inequalities and Lyapunov function approach, the feedback controllers are designed, and some criteria for $\alpha$-synchronization are provided. Finally, the validity of the presented theoretical findings is demonstrated by two specific examples. It is shown that delayed impulses can be viewed as perturbations or stabilizing sources for non-autonomous ICGNNs. Full article

In this paper, the adaptive synchronization problem of quaternion-valued Cohen–Grossberg neural networks (QVCGNNs), with and without known parameters, is investigated. On the basis of constructing an appropriate Lyapunov function, and utilizing parameter identification theory and decomposition methods, two effective adaptive feedback schemes are proposed, to guarantee the realization of global synchronization of CGQVNNs. The control gain of the above schemes can be obtained using the Matlab LMI toolbox. The theoretical results presented in this work enrich the literature exploring the adaptive synchronization problem of quaternion-valued neural networks (QVNNs). Finally, the reliability of the theoretical schemes derived in this work is shown in two interesting numerical examples. Full article

The aim of this article is to develop efficient methods of expressing multilevel structured information from various modalities (images, speech, and text) in order to naturally duplicate the structure as it occurs in the human brain. A number of theoretical and practical issues, including the creation of a mathematical model with a stability point, an algorithm, and software implementation for the processing of offline information; the representation of neural networks; and long-term synchronization of the various modalities, must be resolved in order to achieve the goal. An artificial neural network (ANN) of the Cohen–Grossberg type was used to accomplish the objectives. The research techniques reported herein are based on the theory of pattern recognition, as well as speech, text, and image processing algorithms. Full article