Fixed-time periodic stabilization of discontinuous reaction-diffusion Cohen-Grossberg neural networks.

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction-diffusion Cohen-Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction-diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction-diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Survival analysis and probability density function of switching heroin model.

We study a switching heroin epidemic model in this paper, in which the switching of supply of heroin occurs due to the flowering period and fruiting period of opium poppy plants. Precisely, we give three equations to represent the dynamics of the susceptible, the dynamics of the untreated drug addicts and the dynamics of the drug addicts under treatment, respectively, within a local population, and the coefficients of each equation are functions of Markov chains taking values in a finite state space. The first concern is to prove the existence and uniqueness of a global positive solution to the switching model. Then, the survival dynamics including the extinction and persistence of the untreated drug addicts under some moderate conditions are derived. The corresponding numerical simulations reveal that the densities of sample paths depend on regime switching, and larger intensities of the white noises yield earlier times for extinction of the untreated drug addicts. Especially, when the switching model degenerates to the constant model, we show the existence of the positive equilibrium point under moderate conditions, and we give the expression of the probability density function around the positive equilibrium point.

Topology identification for stochastic multi-layer networks via graph-theoretic method.

The topological structures of multi-layer networks have an important influence on their dynamical properties, but in most cases the topological structures of networks are unknown. Hence, this paper pays attention to investigating topology identification problems for multi-layer networks with stochastic perturbations. Both intra-layer coupling and inter-layer coupling are incorporated into the research model. Based on the graph-theoretic method and Lyapunov function, topology identification criteria for stochastic multi-layer networks are obtained by designing a suitable adaptive controller. Furthermore, to estimate the time of identification, the finite-time identification criteria are obtained by finite-time control technique. Finally, double-layer Watts-Strogatz small-world networks are presented for numerical simulations to illustrate the correctness of theoretical results.

The dynamics of novel corona virus disease via stochastic epidemiological model with vaccination.

During the past two years, the novel coronavirus pandemic has dramatically affected the world by producing 4.8 million deaths. Mathematical modeling is one of the useful mathematical tools which has been used frequently to investigate the dynamics of various infectious diseases. It has been observed that the nature of the novel disease of coronavirus transmission differs everywhere, implying that it is not deterministic while having stochastic nature. In this paper, a stochastic mathematical model has been investigated to study the transmission dynamics of novel coronavirus disease under the effect of fluctuated disease propagation and vaccination because effective vaccination programs and interaction of humans play a significant role in every infectious disease prevention. We develop the epidemic problem by taking into account the extended version of the susceptible-infected-recovered model and with the aid of a stochastic differential equation. We then study the fundamental axioms for existence and uniqueness to show that the problem is mathematically and biologically feasible. The extinction of novel coronavirus and persistency are examined, and sufficient conditions resulted from our investigation. In the end, some graphical representations support the analytical findings and present the effect of vaccination and fluctuated environmental variation.

Outlier-Resistant State Estimation for Singularly Perturbed Complex Networks With Nonhomogeneous Sojourn Probabilities.

This study investigates an outlier-resistant state estimation problem for singularly perturbed complex networks (SPCNs) with sojourn probabilities and randomly occurring coupling strengths. Aiming at better describing the dynamic behavior of the network topology for SPCNs, a novel switching law associated with the time-varying sojourn probabilities is developed, and the variation of sojourn probabilities is arranged by a high-level deterministic switching signal. Meanwhile, a sequence of mode-dependent variables is employed to describe the randomly occurring coupling strength. Subsequently, to alleviate the side effects from possible measurement outliers, a dynamic saturation function-based state estimator is designed, whose saturation level is adaptively varying based on previous estimation errors. In virtue of Lyapunov theory and mode-dependent average dwell-time strategy, it can be verified that the resulting dynamics is stochastic H∞ finite-time bounded. To this end, a simulation example is presented to show the validity of the proposed estimator design method.

New Fixed-Time Stability Criteria of Time-Varying Delayed Discontinuous Systems and Application to Discontinuous Neutral-Type Neural Networks.

This article mainly focuses on putting forward new fixed-time (FIXT) stability lemmas of delayed Filippov discontinuous systems (FDSs). By providing the new inequality conditions imposed on the Lyapunov-Krasovskii functions (LKF), novel FIXT stability lemmas are investigated with the help of inequality techniques. The new settling time is also given and its accuracy is improved in comparison with pioneer ones. For the purpose of illustrating the applicability, a class of discontinuous fuzzy neutral-type neural networks (DFNTNNs) is considered, which includes the previous NTNNs. New criteria are derived and detailed FIXT synchronization results have been obtained. Finally, typical examples are carried out to demonstrate the validity of the main results.

Adaptive Exponential Synchronization of Complex Networks With Nondifferentiable Time-Varying Delay.

In recent years, the adaptive exponential synchronization (AES) problem of delayed complex networks has been extensively studied. Existing results rely heavily on assuming the differentiability of the time-varying delay, which is not easy to verify in reality. Dealing with nondifferentiable delay in the field of AES is still a challenging problem. In this brief, the AES problem of complex networks with general time-varying delay is addressed, especially when the delay is nondifferentiable. A delay differential inequality is proposed to deal with the exponential stability of delayed nonlinear systems, which is more general than the widely used Halanay inequality. Next, the boundedness of the adaptive control gain is theoretically proved, which is neglected in much of the literature. Then, the AES criteria for networks with general delay are established for the first time by using the proposed inequality and the boundedness of the control gain. Finally, an example is given to demonstrate the effectiveness of the theoretical results.

Fixed-Time Stabilization of Discontinuous Neutral Neural Networks With Proportional Delays via New Fixed-Time Stability Lemmas.

When studying the stability of time-delayed discontinuous systems, Lyapunov-Krasovskii functional (LKF) is an essential tool. More relaxed conditions imposed on the LKF are preferred and can take more advantages in real applications. In this article, novel conditions imposed on the LKF are first given which are different from the previous ones. New fixed-time (FXT) stability lemmas are established using some inequality techniques which can greatly extend the pioneers. The new estimations of the settling times (STs) are also obtained. For the purpose of examining the applicability of the new FXT stability lemmas, a class of discontinuous neutral-type neural networks (NTNNs) with proportional delays is formulated which is more generalized than the existing ones. Using differential inclusions theory, set-valued map, and the newly obtained FXT stability lemma, some algebraic FXT stabilization criteria are derived. Finally, examples are given to show the correctness of the established results.

Probing a Stochastic Epidemic Hepatitis C Virus Model with a Chronically Infected Treated Population.

The hepatitis C virus is hitherto a tremendous threat to human beings, but many researchers have analyzed mathematical models for hepatitis C virus transmission dynamics only in the deterministic case. Stochasticity plays an immense role in pathology and epidemiology. Hence, the main theme of this article is to investigate a stochastic epidemic hepatitis C virus model with five states of epidemiological classification: susceptible, acutely infected, chronically infected, recovered or removed and chronically infected, and treated. The stochastic hepatitis C virus model in epidemiology is established based on the environmental influence on individuals, is manifested by stochastic perturbations, and is proportional to each state. We assert that the stochastic HCV model has a unique global positive solution and attains sufficient conditions for the extinction of the hepatotropic RNA virus. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the solutions to the stochastic HCV model. Moreover, this article confirms that using numerical simulations, the six parameters of the stochastic HCV model can have a high impact over the disease transmission dynamics, specifically the disease transmission rate, the rate of chronically infected population, the rate of progression to chronic infection, the treatment failure rate of chronically infected population, the recovery rate from chronic infection and the treatment rate of the chronically infected population. Eventually, numerical simulations validate the effectiveness of our theoretical conclusions.

Design and Analysis of a Novel Distributed Gradient Neural Network for Solving Consensus Problems in a Predefined Time.

In this article, a novel distributed gradient neural network (DGNN) with predefined-time convergence (PTC) is proposed to solve consensus problems widely existing in multiagent systems (MASs). Compared with previous gradient neural networks (GNNs) for optimization and computation, the proposed DGNN model works in a nonfully connected way, in which each neuron only needs the information of neighbor neurons to converge to the equilibrium point. The convergence and asymptotic stability of the DGNN model are proved according to the Lyapunov theory. In addition, based on a relatively loose condition, three novel nonlinear activation functions are designed to speedup the DGNN model to PTC, which is proved by rigorous theory. Computer numerical results further verify the effectiveness, especially the PTC, of the proposed nonlinearly activated DGNN model to solve various consensus problems of MASs. Finally, a practical case of the directional consensus is presented to show the feasibility of the DGNN model and a corresponding connectivity-testing example is given to verify the influence on the convergence speed.

Prefixed-Time Local Intermittent Sampling Synchronization of Stochastic Multicoupling Delay Reaction-Diffusion Dynamic Networks.

This article focuses on the problem of prefixed-time synchronization for stochastic multicoupled delay dynamic networks with reaction-diffusion terms and discontinuous activation by means of local intermittent sampling control. Notably, unlike the existing common fixed-time synchronization, this article puts forward a new synchronization concept, prefixed-time synchronization, based on the fact that stochastic noise and discontinuous activation can be seen everywhere in practical engineering, which can effectively perfect and improve the existing works. Specifically, a local intermittent in the time domain and point sampling control strategy in the spatial domain is proposed instead of a simple single intermittent control approach, which greatly reduces the control cost. In addition, by some effective means, including the famous Young's inequality, Jensen's inequality, and Hölder's inequality, we obtain two different synchronization criteria of the networks without delay and with multicoupling delays and deeply reveal the quantitative relationship among control period, point sampling length, and network scale. Finally, a numerical example is given to verify the effectiveness of the developed method and the practicability by Chua's circuit model.

New Global Asymptotic Robust Stability of Dynamical Delayed Neural Networks via Intervalized Interconnection Matrices.

This article identifies a new upper bound norm for the intervalized interconnection matrices pertaining to delayed dynamical neural networks under the parameter uncertainties. By formulating the appropriate Lyapunov functional and slope-bounded activation functions, the derived new upper bound norms provide new sufficient conditions corresponding to the equilibrium point of the globally asymptotic robust stability with respect to the delayed neural networks. The new upper bound norm also yields the optimized minimum results as compared with some existing methods. Numerical examples are given to demonstrate the effectiveness of the proposed results obtained through the new upper bound norm method.

Intermittent Estimator-Based Mixed Passive and H∞ Control for High-Speed Train With Actuator Stochastic Fault.

This article is concerned with the intermittent estimator-based mixed passive and [Formula: see text] control for the high-speed train (HST) with multiple noises, actuator stochastic fault, and sensor packet loss. First, an intermittent estimator is designed to track the undetectable status of HSTs in response to only partial information available due to sensor failures. Then, two different stability criteria are developed by adopting two different Lyapunov function strategies. Simultaneously, in order to reduce the control cost and accelerate the convergence time, two different algorithms are designed. It is worth emphasizing that different from the existing results of HST subject to actuator fault, this article adopts a more flexible fault representation mode, namely, semi-Markov switching mode, which is more in line with the practical background and has a higher valuable application. Especially, the Lyapunov function designed in this article can drive the system state to decrease monotonically in both the "working interval" and the "rest interval," so as to avoid the phenomenon of state impulsive jump. Finally, through the test of HST experimental value of Japan's Shinkansen, the simulation results show the effectiveness and rationality of the proposed control method and also make a comparative analysis with related works, to prove the advantages of the control technology proposed in this article.

Performance Analysis and Applications of Finite-Time ZNN Models With Constant/Fuzzy Parameters for TVQPEI.

Based on extensive applications of the time-variant quadratic programming with equality and inequality constraints (TVQPEI) problem and the effectiveness of the zeroing neural network (ZNN) to address time-variant problems, this article proposes a novel finite-time ZNN (FT-ZNN) model with a combined activation function, aimed at providing a superior efficient neurodynamic method to solve the TVQPEI problem. The remarkable properties of the FT-ZNN model are faster finite-time convergence and preferable robustness, which are analyzed in detail, where in the case of the robustness discussion, two kinds of noises (i.e., bounded constant noise and bounded time-variant noise) are taken into account. Moreover, the proposed several theorems all compute the convergent time of the nondisturbed FT-ZNN model and the disturbed FT-ZNN model approaching to the upper bound of residual error. Besides, to enhance the performance of the FT-ZNN model, a fuzzy finite-time ZNN (FFT-ZNN), which possesses a fuzzy parameter, is further presented for solving the TVQPEI problem. A simulative example about the FT-ZNN and FFT-ZNN models solving the TVQPEI problem is given, and the experimental results expectably conform to the theoretical analysis. In addition, the designed FT-ZNN model is effectually applied to the repetitive motion of the three-link redundant robot and image fusion to show its potential practical value.

New Fixed-Time Stability Analysis of Delayed Discontinuous Systems via an Augmented Indefinite Lyapunov-Krasovskii Functional.

This article discusses the fixed-time stability (FTS) of a kind of delayed discontinuous system (DS) in Filippov sense. Based on the set-valued map, the FTS analysis of the general solution is first transformed into the zero solution of the differential inclusion. Second, the new criteria of the Lyapunov-Krasovskii functional (LKF) are given and LKF is proved to possess the indefinite derivatives by using the simple integral inequalities. In addition, the FTS of the considered delayed DS is achieved and the new settling time is estimated. Third, to demonstrate the applicability of the new FTS theorems, the FTS control of a class of discontinuous inertial neural networks (DINNs) with time-varying delays is solved. Finally, two numerical examples are given to examine the theoretical results and simulations are also provided to make some illustrations.

Exponential State Estimation for Memristor-Based Discrete-Time BAM Neural Networks With Additive Delay Components.

This paper focuses on the dynamical behavior for a class of memristor-based bidirectional associative memory neural networks (BAMNNs) with additive time-varying delays in discrete-time case. The necessity of the proposed problem is to design a proper state estimator such that the dynamics of the corresponding estimation error is exponentially stable with a prescribed decay rate. By constructing an appropriate Lyapunov-Krasovskii functional (LKF) and utilizing Cauchy-Schwartz-based summation inequality, the delay-dependent sufficient conditions for the existence of the desired estimator are derived in the absence of uncertainties which are further extended to available uncertain parameters of the prescribed memristor-based BAMNNs in terms of linear matrix inequalities (LMIs). By solving the proposed LMI conditions the estimation gain matrices are obtained. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed results.

Dynamical Behavior of Nonautonomous Stochastic Reaction-Diffusion Neural-Network Models.

This brief investigates nonautonomous stochastic reaction-diffusion neural-network models with S-type distributed delays. First, the existence and uniqueness of mild solution are studied under the Lipschitz condition without the linear growth condition. Due to the existence of a nonautonomous reaction-diffusion term and the infinite dimensional Wiener process, the criteria for the well-posedness of the models are established based on the evolution system theory. Then, the S-type distributed delay, which is an infinite delay, is handled by the truncation method, and sufficient conditions for the global exponential stability are obtained by constructing a simple Lyapunov-Krasovskii functional candidate. Finally, neural-network examples and an illustrative example are given to show the applications of the obtained results.

An Improved Result on Dissipativity and Passivity Analysis of Markovian Jump Stochastic Neural Networks With Two Delay Components.

In this paper, we investigate the dissipativity and passivity of Markovian jump stochastic neural networks involving two additive time-varying delays. Using a Lyapunov-Krasovskii functional with triple and quadruple integral terms, we obtain delay-dependent passivity and dissipativity criteria for the system. Using a generalized Finsler lemma (GFL), a set of slack variables with special structure are introduced to reduce design conservatism. The dissipativity and passivity criteria depend on the upper bounds of the discrete time-varying delay and its derivative are given in terms of linear matrix inequalities, which can be efficiently solved through the standard numerical software. Finally, our illustrative examples show that the proposed method performs well and is successful in problems where existing methods fail.

Mean-square exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching based on vector Lyapunov functions.

This paper studies the mean-square exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. By using the vector Lyapunov function and property of M-matrix, two generalized Halanay inequalities are established. By means of the generalized Halanay inequalities, sufficient conditions are also obtained, which can ensure the exponential input-to-state stability of delayed Cohen-Grossberg neural networks with Markovian switching. Two numerical examples are given to illustrate the efficiency of the derived results.

New stability criterion of neural networks with leakage delays and impulses: a piecewise delay method.

This paper analyzes the global asymptotic stability of a class of neural networks with time delay in the leakage term and time-varying delays under impulsive perturbations. Here the time-varying delays are assumed to be piecewise. In this method, the interval of the variation is divided into two subintervals by its central point. By developing a new Lyapunov-Krasovskii functional and checking its variation in between the two subintervals, respectively, and then we present some sufficient conditions to guarantee the global asymptotic stability of the equilibrium point for the considered neural network. The proposed results which do not require the boundedness, differentiability and monotonicity of the activation functions, can be easily verified via the linear matrix inequality (LMI) control toolbox in MATLAB. Finally, a numerical example and its simulation are given to show the conditions obtained are new and less conservative than some existing ones in the literature.