99 research outputs found

    Robustness analysis of Cohen-Grossberg neural network with piecewise constant argument and stochastic disturbances

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    Robustness of neural networks has been a hot topic in recent years. This paper mainly studies the robustness of the global exponential stability of Cohen-Grossberg neural networks with a piecewise constant argument and stochastic disturbances, and discusses the problem of whether the Cohen-Grossberg neural networks can still maintain global exponential stability under the perturbation of the piecewise constant argument and stochastic disturbances. By using stochastic analysis theory and inequality techniques, the interval length of the piecewise constant argument and the upper bound of the noise intensity are derived by solving transcendental equations. In the end, we offer several examples to illustrate the efficacy of the findings

    Exponential Lag Synchronization of Cohen-Grossberg Neural Networks with Discrete and Distributed Delays on Time Scales

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    In this article, we investigate exponential lag synchronization results for the Cohen-Grossberg neural networks (C-GNNs) with discrete and distributed delays on an arbitrary time domain by applying feedback control. We formulate the problem by using the time scales theory so that the results can be applied to any uniform or non-uniform time domains. Also, we provide a comparison of results that shows that obtained results are unified and generalize the existing results. Mainly, we use the unified matrix-measure theory and Halanay inequality to establish these results. In the last section, we provide two simulated examples for different time domains to show the effectiveness and generality of the obtained analytical results.Comment: 20 pages, 18 figure

    Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations

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    This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper

    Nonlinear Systems

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    Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

    Deep learning applied to computational mechanics: A comprehensive review, state of the art, and the classics

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    Three recent breakthroughs due to AI in arts and science serve as motivation: An award winning digital image, protein folding, fast matrix multiplication. Many recent developments in artificial neural networks, particularly deep learning (DL), applied and relevant to computational mechanics (solid, fluids, finite-element technology) are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. Hybrid methods combine traditional PDE discretizations with ML methods either (1) to help model complex nonlinear constitutive relations, (2) to nonlinearly reduce the model order for efficient simulation (turbulence), or (3) to accelerate the simulation by predicting certain components in the traditional integration methods. Here, methods (1) and (2) relied on Long-Short-Term Memory (LSTM) architecture, with method (3) relying on convolutional neural networks. Pure ML methods to solve (nonlinear) PDEs are represented by Physics-Informed Neural network (PINN) methods, which could be combined with attention mechanism to address discontinuous solutions. Both LSTM and attention architectures, together with modern and generalized classic optimizers to include stochasticity for DL networks, are extensively reviewed. Kernel machines, including Gaussian processes, are provided to sufficient depth for more advanced works such as shallow networks with infinite width. Not only addressing experts, readers are assumed familiar with computational mechanics, but not with DL, whose concepts and applications are built up from the basics, aiming at bringing first-time learners quickly to the forefront of research. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics, even in well-known references. Positioning and pointing control of a large-deformable beam is given as an example.Comment: 275 pages, 158 figures. Appeared online on 2023.03.01 at CMES-Computer Modeling in Engineering & Science

    Controlling stability through the rate of decay of the delay feedback kernels

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    Of concern is the Hopfield neural network system comprising discrete as well as distributed delays in the form of a convolution. For a desired convergence rate of the solution to the equilibrium state, we establish sufficient conditions on the delay kernels ensuring this matter. Our result improves an existing one in the literature. The adopted approach is completely different. It relies on a judicious choice of a Lyapunov-like function and careful manipulations

    Robust stability and boundedness of uncertain conformable fractional-order delay systems under input saturation

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    In this article, a class of uncertain conformable fractional-order delay systems under input saturation is considered. By establishing the Lyapunov boundedness theorem for conformable fractional-order delay systems, some sufficient conditions for robust stability and boundedness of the systems are obtained. Examples are given to illustrate the obtained theory

    New criteria on global Mittag-Leffler synchronization for Caputo-type delayed Cohen-Grossberg Inertial Neural Networks

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    Our focus of this paper is on global Mittag-Leffler synchronization (GMLS) of the Caputo-type Inertial Cohen-Grossberg Neural Networks (ICGNNs) with discrete and distributed delays. This model takes into account the inertial term as well as the two types of delays, which greatly reduces the conservatism with respect to the model. A change of variables transforms the 2β 2\beta order inertial frame into β \beta order ordinary frame in order to deal with the effect of the inertial term. In the following steps, two novel types of delay controllers are designed for the purpose of reaching the GMLS. In conjunction with the novel controllers, utilizing differential mean-value theorem and inequality techniques, several criteria are derived to determine the GMLS of ICGNNs within the framework of Caputo-type derivative and calculus properties. At length, the feasibility of the results is further demonstrated by two simulation examples

    Fractional Calculus - Theory and Applications

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    In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

    Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

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    Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality, Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab
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