Robust convergence of Cohen–Grossberg neural networks with mode-dependent time-varying delays and Markovian jump

The robust stochastic convergence in mean square is investigated for a class of uncertain Cohen–Grossberg neural networks with both Markovian jump parameters and mode-dependent time-varying delays. By employing the Lyapunov method and a generalized Halanay-type inequality, a delay-dependent condition is derived to guarantee the state variables of the discussed neural networks to be globally uniformly exponentially stochastic convergent to a ball in the state space with a pre-specified convergence rate. After some parameters being fixed in advance, the proposed conditions are all in terms of linear matrix inequalities, which can be solved numerically by employing the LMI toolbox in Matlab. Finally, an illustrated example is given to show the effectiveness and usefulness of the obtained results.     

Global exponential stability of discrete-time higher-order Cohen–Grossberg neural networks with time-varying delays,connection weights and impulses

In this paper, an auxiliary model-based nonsingular M-matrix approach is used to establish the global exponential stability of the zero equilibrium, for a class of discrete-time high-order Cohen–Grossberg neural networks (HOCGNNs) with time-varying delays, connection weights and impulses. A new impulse-free discrete-time HOCGNN with time-varying delays and connection weights is firstly constructed, and the relationship between the solutions of original systems and new HOCGNNs is indicated by a technical lemma. From which, the global exponential stability criteria for the zero equilibrium are derived by using an inductive idea and the properties of nonsingular M-matrices. The effectiveness of the obtained stability criteria is illustrated by numerical examples. Compared with the previous results, this paper has three advantages: (i) The Lyapunov–Krasovskii functional is not required; (ii) The obtained global exponential stability criteria are applied to check whether a matrix is a nonsingular M-matrix, which can be conveniently tested; (iii) The proposed approach applies to most of discrete-time system models with impulses and delays.     

Exponential synchronization in inertial Cohen–Grossberg neural networks with time delays

In this paper, the problem of exponential synchronization for inertial Cohen–Grossberg neural networks with time delays is studied. According to the concept of synchronization, a controlled response system is constructed to obtain the error systems. First, by introducing a directive Lyapunov functional, a sufficient condition is derived to ascertain the exponential synchronization of the drive and response systems based on feedback control. Moreover, by introducing a variable substitution, a sufficient condition is obtained to ensure the global exponential synchronization for the systems. Two sufficient conditions are feasible for the global exponential synchronization of the drive and response systems, and complement each other. Finally, the parameters were set for numerical simulation, two illustrative examples are provided to show the effectiveness of the obtained theoretical results, and the validity of the model was proved.     

Finite-time synchronization of coupled Cohen–Grossberg neural networks with and without coupling delays

This paper analyzes synchronization in finite time for two types of coupled delayed Cohen–Grossberg neural networks (CDCGNNs). In the first type, linearly coupled Cohen–Grossberg neural networks with and without coupling delays are considered, respectively. In the second type, nonlinearly coupled Cohen–Grossberg neural networks both with and without coupling delays are discussed. By designing suitable controllers and using some inequality techniques, several criteria ensuring finite-time synchronization of the CDCGNNs with linear coupling and nonlinear coupling are derived, respectively. Moreover, the settling times of synchronization in finite time for the considered networks are also predicted. In the end, the availability for the acquired finite-time synchronization conditions is confirmed by two selected numerical examples.     

Passivity and robust passivity of stochastic reaction–diffusion neural networks with time-varying delays

In this paper, passivity and robust passivity for a general class of stochastic reaction–diffusion neural networks with Dirichlet boundary conditions and discrete time-varying delays are considered. With the help of inequality techniques and stochastic analysis, sufficient conditions are developed to guarantee passivity and robust passivity of the addressed neural networks. The obtained results in this study include some existing ones as special cases. A numerical example is carried out to illustrate the feasibility of the proposed theoretical criteria.     

Passivity analysis of coupled neural networks with reaction–diffusion terms and mixed delays

In this paper, we intend to discuss the passivity of coupled neural networks (NNs) with reaction–diffusion terms and mixed delays. By constructing appropriate Lyapunov functional, and with the help of liner matrix inequalities, some inequality techniques, several sufficient conditions are derived to guarantee the output strictly passive, input strictly passive, passive of the proposed neural network model. Then, a stability criterion is presented according to the obtained passivity results. Moreover, the proposed neural network model herein is more general than some recent studies, which can improve and enrich the previous research results. Finally, a numerical example is presented to show the effectiveness of the theoretical criteria.     

Robust synchronization of discontinuous Cohen–Grossberg neural networks: Pinning control approach

In this issue, the robust synchronization for a class of uncertain Cohen–Grossberg neural networks is studied, in which neuron activations are modelled by discontinuous functions(or piecewise continuous functions). Pinning state-feedback and adaptive controllers are designed to achieve global robust exponential synchronization and global robust asymptotical synchronization of drive-response-based discontinuous Cohen–Grossberg neural networks. By applying the theory of non-smooth analysis theory and the method of generalized Lyapunov functional, some criteria are given to show that the coupled discontinuous Cohen–Grossberg neural networks with parameter uncertainties can realized global robust synchronization. Some examples and numerical simulations are also shown to verify the validity of the proposed results.     

New robust H∞ control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method

This paper considers the problems of robust stochastic stabilization and robust $H_{\infty }$ controller design for a class of stochastic Markovain jumping systems with mixed time delays and polytopic parameter uncertainties. Both the interval time-varying delay and distributed time delay are simultaneously considered. Some new delay-dependent sufficient conditions, which differs greatly from the most existing results, are obtained based on the decoupling method and some advanced techniques. A numerical example is provided to illustrate the effectiveness of the proposed criteria.     

Finite-time synchronization for periodic T–S fuzzy master-slave neural networks with distributed delays

Finite-time (FT) synchronization for periodic T–S fuzzy master-slave neural networks (NNs) with distributed delays is addressed in this work. A fuzzy controller is designed for the salve NNs to synchronize the master NNs in FT and a synchronization error system (SES) is derived. Sufficient conditions are established to guarantee that the SES is FT bounded by using the mode and fuzzy basis dependent Lyapunov function. A new algorithm is proposed to obtain the suboptimal boundary of the SES to analyze how the periodic characteristics affect the system boundary. Finally, a numerical example is provided to demonstrate the validity of the fuzzy controller and the iterative algorithm for the boundary.     

Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction–diffusion terms via sampled-data control

This paper is concerned with the problem of robust synchronization of a class of complex dynamical networks with time-varying delays and reaction–diffusion terms. To reflect most of the dynamical behaviors of the system, the parameter uncertainties are considered. A sampled-data controller with m stochastically varying sampling periods whose occurrence probabilities are given constants is considered. The control objective is that the trajectories of the system by designing suitable control schemes track the trajectories of the system with sample-data control. It is shown that, through Lyapunov stability theory, the proposed sample-data controllers are successful in ensuring the achievement of robust synchronization of complex dynamical networks even in the case of uncertainity and Markovian jumping parameters. By utilizing the Lyapunov functional method, Jensen’s inequality, Wirtinger’s inequality and lower bounds theorem, we establish a sufficient criterion such that, for all admissible parameter uncertainties, the complex dynamical network is robustly synchronized. The derived criteria are expressed in terms of linear matrix inequalities that can be easily checked by using the standard numerical software. Illustrative examples are presented to demonstrate the effectiveness and usefulness of the proposed results.     

A state estimation H∞ issue for discrete-time stochastic impulsive genetic regulatory networks in the presence of leakage,multiple delays and Markovian jumping parameters

In this work, we probes the stability results of H_∞ state estimation for discrete-time stochastic genetic regulatory networks with leakage, distributed delays, Markovian jumping parameters and impulsive effects. Here, we focus to evaluate the true absorption of mRNAs and proteins by calculating the H_∞ estimator in such a way that the estimation error dynamics is stochastically stable during the completion of the prescribed H_∞ disturbance attenuation level. In favor of decreasing the data communion in trouble, the H_∞ system accept and evaluate the outputs that are only transferred to the estimator when a certain case is acroses. Further, few sufficient conditions are formulated, by utilizing the Lyapunov–Krasovskii functional under which the estimation error system is stochastically stable and also satisfied the H_∞ attainment constraint. The estimator is obtained in terms of linear matrix inequalities (LMIs) and these LMIs are attainable, only if the estimator gains can be absolutely given. In addition to that, two numerical examples are exposed to establish the efficiency of our obtained results.     

A sliding mode approach to H∞ synchronization of master–slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties

In this paper, a sliding-mode approach is proposed for exponential H_∞ synchronization problem of a class of master–slave time-delay systems with both discrete and distributed time-delays, norm-bounded nonlinear uncertainties and Markovian switching parameters. Using an appropriate Lyapunov–Krasovskii functional, some delay-dependent sufficient conditions and a synchronization law, which include the master–slave parameters are established for designing a delay-dependent mode-dependent sliding mode exponential H_∞ synchronization control law in terms of linear matrix inequalities. The controller guarantees the H_∞ synchronization of the two coupled master and slave systems regardless of their initial states. Two numerical examples are given to show the effectiveness of the method.     

Stability in mean of partial variables for stochastic reaction–diffusion systems with Markovian switching

This paper is denoted to investigating stability in mean of partial variables for stochastic reaction–diffusion equations with Markovian switching (SRDEMS). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations with Markovian switching (SODEMS) and using Itô formula, sufficient criteria on uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, exponential stability in mean of partial variables for SRDEMS are first derived. An example is presented to illustrate the effectiveness and efficiency of the obtained results.     

Stabilization of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays and bounded controls

This paper deals with the problem of stabilization by state feedback control of Takagi–Sugeno (T–S) fuzzy discrete-time systems with multiple fixed delays while imposing positivity in closed-loop. The obtained results are presented under linear programming (LP) form. In particular, the synthesis of state feedback controllers is first solved in terms of Linear programming for the unbounded controls case. This result is then extended to the stabilization problem by nonnegative controls, and stabilization by bounded controls. The stabilization conditions are derived using the single Lyapunov–Krasovskii functional (LKF). An example of a real plant is studied to show the advantages of the design procedure. A comparison between linear programming and LMI approaches is presented.     

Global mean square exponential stability and stabilization of uncertain switched delay systems with Lévy noise and flexible switching signals

This paper focuses on the stability analysis and stabilization problem for a class of uncertain switched delay systems with Lévy noise and flexible switching signals which unify the high-frequency switching and low-frequency switching. By employing the theory of switched systems, mathematical induction and stochastic analysis technique, some sufficient conditions in form of algebraic inequalities are derived to guarantee the stability and stabilization of such systems. Different from dwell time and average dwell time, the proposed switching rule constrained the partial dwell-time shows that the switching number in the same time interval can be more elastic. Finally, numerical examples are implemented to illustrate the effectiveness of the theoretical results.     

Quadratically convex combination approach to stability of T–S fuzzy systems with time-varying delay

This paper develops a novel stability analysis method for Takagi–Sugeno (T–S) fuzzy systems with time-varying delay. New delay-dependent stability criteria in terms of linear matrix inequalities for time-varying delayed T–S fuzzy systems are derived by the newly proposed augmented Lyapunov–Krasovski (L–K) functional. This functional contains the cross terms of variables and quadratic terms multiplied by a higher degree scalar function. Different from previous results, our derivation applies the idea of second-order convex combination, and the property of quadratic convex function without resorting to the Jensen's inequality. Two numerical examples are provided to verify the effectiveness of the presented results.     

Novel master–slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control

This paper investigates the problem of master–slave synchronization of chaotic Lur’e systems (CLSs) with time delays by sampled-data control. First, a novel Lyapunov–Krasovskii functional (LKF) is constructed with some new augmented terms, which can fully capture the system characteristics and the available information on the actual sampling pattern. In comparison with existing results, the constraint condition of the positive definition of the LKF is more relax, since it is positive definite only requiring at sampling instants. Second, based on the LKF, a less conservative synchronization criterion is established. Third, the desired estimator gain can be designed in terms of the solution to linear matrix inequalities (LMIs). The obtained conditions ensure the master–slave synchronization of CLSs under a longer sampling period than remarkable existing works. Finally, three numerical simulations of Chua’s circuit and neural network are provided to show the effectiveness and advantages of the proposed results.     

Sampled-data H∞ filtering of Takagi–Sugeno fuzzy systems with interval time-varying delays

The sampled-data _H∞$H_{\infty }$ filtering for a continuous-time Takagi–Sugeno fuzzy system with an interval time-varying state delay is investigated, where the measurement outputs from the plant to the filter are assumed to be sampled at discrete instants with a variable period. Firstly, by means of a newly proposed inequality bounding technique and a new Lyapunov–Krasovskii functional, the fuzzy sampled-data _H∞$H_{\infty }$ filtering performance analysis is carried out such that the resultant filter error system is asymptotically stable with a prescribed _H∞$H_{\infty }$ attenuation performance index. Secondly, sufficient conditions on the existence of fuzzy sampled-data _H∞$H_{\infty }$ filters are derived in the simultaneous presence of the time-varying state delay and the variable sampling period. The proposed bounding inequality lies in its more tightness and alleviates the enlargement of some inverse “coefficients” resulting from the utilization of the well-known Jensen integral inequality  . Compared with some existing Lyapunov–Krasovskii functionals, more information about the relationship among the current state and its delayed state is considered. The upper bound of the derivative of the time-varying state delay is not required to be less than one. Different from some existing results in the literature, by applying the proposed results, each different value of such an upper bound (greater than one) leads to a different _H∞$H_{\infty }$ disturbance attenuation level. Finally, a numerical example and a modified continuous stirred tank reactor system are given to show the effectiveness of the proposed results.     

Hierarchical type stability and stabilization of networked control systems with event-triggered mechanism via canonical Bessel–Legendre inequalities

This paper is studied with the hierarchical type stability and stabilization of networked control systems (NCSs) with event-triggered mechanism (ETM). In the cause of reducing the amount of data transmission and saving the limited network bandwidth, ETM is introduced into NCSs, and the closed-loop time-delay NCSs model with ETM is presented. An improved Lyapunov–Krasovskii functional (LKF), containing delay-product-type terms and being appropriate for the canonical BesselLegendre inequality (BLI), is first constructed. Then, by utilizing the canonical BLI and the extended reciprocally convex matrix inequality (ERCMI) to deal with the single integral terms of the derivative of LKF, a sufficient condition on asymptotically stable is derived for NCSs. Based on above N-dependent stability criteria, a co-design method is developed, which can be capable of calculating the control gain of controller and the weighting matrix of the ETM. Finally, the feasibility and superiority of the results are verified by two examples.