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Stochastic Differential Equations And Applications

This brief is concerned with the stability criteria for recurrent neural networks with time-varying delay. First, based on convex combination technique, a delay interval with fixed terminals is changed into the one with flexible terminals, which is called flexible terminal method (FTM). Second, based on the FTM, a novel Lyapunov–Krasovskii functional is constructed, in which the integral interval associated with delayed variables is not fixed. Thus, the FTM can achieve the same effect as that of delay-partitioning method, while their implementary ways are different. Guided by FTM, Wirtinger-based integral inequality and free-weight matrix method are employed to develop several stability criteria, respectively. Finally, the feasibility and the effectiveness of the proposed results are tested by two numerical examples.

Stability of Recurrent Neural Networks With Time-Varying Delay via Flexible Terminal Method

Fixed-time synchronization of quaternion-valued memristive neural networks with time delays.

The finite-time Mittag-Leffler stability for fractional-order quaternion-valued memristive neural networks (FQMNNs) with impulsive effect is studied here. A new mathematical expression of the quaternion-value memductance (memristance) is proposed according to the feature of the quaternion-valued memristive and a new class of FQMNNs is designed. In quaternion field, by using the framework of Filippov solutions as well as differential inclusion theoretical analysis, suitable Lyapunov-functional and some fractional inequality techniques, the existence of unique equilibrium point and Mittag-Leffler stability in finite time analysis for considered impulsive FQMNNs have been established with the order $$0<\beta <1$$. Then, for the fractional order $$\beta $$ satisfying $$1<\beta <2$$ and by ignoring the impulsive effects, a new sufficient criterion are given to ensure the finite time stability of considered new FQMNNs system by the employment of Laplace transform, Mittag-Leffler function and generalized Gronwall inequality. Furthermore, the asymptotic stability of such system with order $$1<\beta <2$$ have been investigated. Ultimately, the accuracy and validity of obtained finite time stability criteria are supported by two numerical examples.

Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses

This paper examines the problem of asymptotic stability criteria for Markovian jump generalized neural networks with successive time-varying delay components. Generalized neural networks consist of a finite number of modes, which may jump from one mode to another according to a Markovian chain with known transition probability. By constructing novel augmented Lyapunov–Krasovskii functionals (LKFs) with triple integral terms that contain more and more information on the state vectors of the NNs, the upper bound of the successive time-varying delays is formulated. By employing a new integral inequality technique, free-weighting matrix-based integral inequality approach, and Wirtinger double integral inequality technique and that is combined with the reciprocally convex combination approach to estimate the single and double integral terms in the time derivative of the LKFs, a new set of delay-dependent conditions for the asymptotic stability of the considered NNs are represented in the form of linear matrix inequalities. Finally, five numerical examples are given to verify the effectiveness of the proposed approach with a four-tank benchmark real-world problem.

New Criteria for Stability of Generalized Neural Networks Including Markov Jump Parameters and Additive Time Delays

Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function

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.

https://www.mdpi.com/2227-7390/8/3/422/pdf

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

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.

https://www.mdpi.com/2227-7390/8/5/742/pdf

Robust Stability of Complex-Valued Stochastic Neural Networks with Time-Varying Delays and Parameter Uncertainties

In this article, we investigate the robust passivity and stability analysis of uncertain complex-valued impulsive neural network (UCVINN) models with time-varying delays. Many practical systems are subject to uncertainty in the real-world environments. As a result, we consider the uncertainty of norm-bounded parameters to achieve more realistic system behaviors. By using appropriate Lyapunov–Krasovskii functionals and integral inequalities, sufficient conditions for the robust passivity and global asymptotic stability of UCVINNs are derived by separating complex-valued neural networks into real and imaginary parts. The criteria are given in terms of linear matrix inequalities (LMIs) that can be checked by the MATLAB LMI toolbox. Finally, numerical simulations are presented to illustrate the merits of the obtained results.

Ramalingam Sriraman

Papers

Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function

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

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

Robust Stability of Complex-Valued Stochastic Neural Networks with Time-Varying Delays and Parameter Uncertainties

Robust Passivity and Stability Analysis of Uncertain Complex-Valued Impulsive Neural Networks with Time-Varying Delays