Abstract:
This paper is devoted to the study of the stochastic stability of a class of Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying. With the help of Lyapunov function, Burkholder-Davids-Gundy inequality, and Borel-Cantell's theory, a set of novel sufficient conditions on th moment exponential stability and almost sure exponential stability for the trivial solution of the system is derived. Compared with the previous published results, our method does not resort to the Razumikhin-type theorem and the semimartingale convergence theorem. Results of the development as presented in this paper are more general than those reported in some previously published papers. An illustrative example is also given to show the effectiveness of the obtained results.

Abstract:
This paper studies the global synchronization problem for a class of complex networks with discrete time delays. By using the theory of calculus on time scales, the properties of Kronecker product, and Lyapunov method, some sufficient conditions are obtained to ensure the global synchronization of the complex networks with delays on time scales. These sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). The main contribution of the result is that the global synchronization problems with both discrete time and continuous time are unified under the same framework. 1. Introduction As well known, complex dynamical networks have been a subject of high importance and increasing interest within the science and technology communities. The synchronization is one of the most typical phenomena in complex networks, which is ubiquitous in the real world, such as secure communication, chaos generators design, and harmonic oscillation generation, ([1–8], and references cited therein). During the past many years, the synchronization of complex networks has received increasing research attention. There are lots of the papers studying the continuous time and the discrete time dynamical systems. However, most of the investigations are restricted to the continuous or discrete systems, respectively, [9–22]. For avoiding this trouble, it is meaningful to study this problem on time scales which can unify the continuous and discrete dynamical systems under the unified framework. The theory of time scale calculus was initiated by Hilger in 1988, developed, and consummated by Bohner and Peterson [23–25], which has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, population dynamics, biotechnology, economics, and so on [26, 27]. This novel and fascinating type of mathematics is more general and versatile than the traditional theories of differential and difference equations as it can, under one framework, mathematically describe continuous and discrete hybrid processes and hence is the optimal way forward for accurate and malleable mathematical modeling. The field of dynamic equations on time scales contains, links, and extends the classical theory of differential and difference equations. However, to the best of our knowledge, there are few works investigating the synchronization problem of complex networks with delays on time scales. Notations. Throughout this paper, and denote the n-dimensional Euclidean space and the set of all real matrices, respectively. is a time scale,

Abstract:
The problem of stochastic stability is investigated for a class of neural networks with both Markovian jump parameters and continuously distributed delays. The jumping parameters are modeled as a continuous-time, finite-state Markov chain. By constructing appropriate Lyapunov-Krasovskii functionals, some novel stability conditions are obtained in terms of linear matrix inequalities (LMIs). The proposed LMI-based criteria are computationally efficient as they can be easily checked by using recently developed algorithms in solving LMIs. A numerical example is provided to show the effectiveness of the theoretical results and demonstrate the LMI criteria existed in the earlier literature fail. The results obtained in this paper improve and generalize those given in the previous literature.

Abstract:
The problems on global dissipativity and global exponential dissipativity are investigated for uncertain discrete-time neural networks with time-varying delays and general activation functions. By constructing appropriate Lyapunov-Krasovskii functionals and employing linear matrix inequality technique, several new delay-dependent criteria for checking the global dissipativity and global exponential dissipativity of the addressed neural networks are established in linear matrix inequality (LMI), which can be checked numerically using the effective LMI toolbox in MATLAB. Illustrated examples are given to show the effectiveness of the proposed criteria. It is noteworthy that because neither model transformation nor free-weighting matrices are employed to deal with cross terms in the derivation of the dissipativity criteria, the obtained results are less conservative and more computationally efficient. 1. Introduction In the past few decades, delayed neural networks have found successful applications in many areas such as signal processing, pattern recognition, associative memories, parallel computation, and optimization solvers [1]. In such applications, the qualitative analysis of the dynamical behaviors is a necessary step for the practical design of neural networks [2]. Many important results on the dynamical behaviors have been reported for delayed neural networks; see [1–16] and the references therein for some recent publications. It should be pointed out that all of the abovementioned literatures on the dynamical behaviors of delayed neural networks are concerned with continuous-time case. However, when implementing the continuous-time delayed neural network for computer simulation, it becomes essential to formulate a discrete-time system that is an analogue of the continuous-time delayed neural network. To some extent, the discrete-time analogue inherits the dynamical characteristics of the continuous-time delayed neural network under mild or no restriction on the discretization step-size, and also remains some functional similarity [17]. Unfortunately, as pointed out in [18], the discretization cannot preserve the dynamics of the continuous-time counterpart even for a small sampling period, and therefore there is a crucial need to study the dynamics of discrete-time neural networks. Recently, the dynamics analysis problem for discrete-time delayed neural networks and discrete-time systems with time-varying state delay has been extensively studied; see [17–21] and references therein. It is well known that the stability problem is central to the

Abstract:
This paper is devoted to exponential synchronization for complex dynamical networks with delay and impulsive effects. The coupling configuration matrix is assumed to be irreducible. By using impulsive differential inequality and the Kronecker product techniques, some criteria are obtained to guarantee the exponential synchronization for dynamical networks. We also extend the delay fractioning approach to the dynamical networks by constructing a Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical examples are given to demonstrate the theoretical results.

Abstract:
The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results. 1. Introduction Recently, synchronization of all dynamical nodes in a network is one of the hot topics in the investigation of complex networks. It is well known that there are many useful synchronization phenomena in real life, such as the synchronous transfer of digital or analog signals in communication networks. Adaptive feedback control has witnessed its effectiveness in synchronizing a complex network [1–4]. By using the adaptive feedback control scheme, Chen and Zhou [1] studied synchronization of complex nondelayed networks, Cao et al. [2] investigated the complete synchronization in an array of linearly stochastically coupled identical networks with delays. In [3], Zhou et al. considered complex dynamical networks with uncertain couplings. In [4], Lu et al. studied the synchronization in arrays of delay-coupled neural networks. However, it is assumed that all the nodes need to be controlled in [1–4]. As we know, the real-world complex networks normally have a large number of nodes; it is usually impractical and impossible to control a complex network by adding the controllers to all nodes. To overcome this difficulty, pinning control, in which controllers are only applied to a small fraction of nodes, has been introduced in recent years [5–9]. By using adaptive pinning control method, Zhou et al. [10] studied local and global synchronizations of complex networks without delays; authors of [11, 12] considered the global synchronizations of the complex networks with nondelayed and delayed couplings. Note that, in most of existing results of complex networks' synchronization, all of the states are coupled for connected nodes. However, adaptive pinning synchronization results in which only partial states of the nodes are coupled are few. Hence, in this paper, we consider two different adaptive pinning controllers, which synchronize complex networks with partial or complete couplings of the nodes' states. In the process of studying synchronization of complex networks, two

Abstract:
By using a Lyapunov-Krasovskii functional method and the stochastic analysis technique, we investigate the problem of synchronization for discrete-time stochastic neural networks (DSNNs) with random delays. A control law is designed, and sufficient conditions are established that guarantee the synchronization of two identical DSNNs with random delays. Compared with the previous works, the time delay is assumed to be existent in a random fashion. The stochastic disturbances are described in terms of a Brownian motion and the time-varying delay is characterized by introducing a Bernoulli stochastic variable. Two examples are given to illustrate the effectiveness of the proposed results. The main contribution of this paper is that the obtained results are dependent on not only the bound but also the distribution probability of the time delay. Moreover, our results provide a larger allowance variation range of the delay, and are less conservative than the traditional delay-independent ones.

Abstract:
Without assuming the symmetry and irreducibility of the outer-coupling weight configuration matrices, we investigate the pinning synchronization of delayed neural networks with nonlinear inner-coupling. Some delay-dependent controlled stability criteria in terms of linear matrix inequality (LMI) are obtained. An example is presented to show the application of the criteria obtained in this paper. 1. Introduction and Model Description In the past few decades, the problem of control and synchronization in complex networks has attracted increasing attention. There are attempts to control the dynamics of a complex network and guide it to a desired state, such as an equilibrium point or a periodic orbit of the network. Since a complex network has a large number of nodes, it is difficult to control it by adding controllers to all nodes. To reduce the number of the controllers, Wang and Chen investigated pinning control for complex networks [1]. Pinning control applies local feedback injections to a small fraction of nodes on a large-size network, thereby achieving some intended global performances over the entire network. In [1], Wang and Chen showed that, due to the extremely inhomogeneous connectivity distribution of scale-free networks, it is much effective to pin some most-highly connected nodes than to pin randomly selected nodes. In [2], Li et al. further investigated the control of complete random networks and scale-free networks via virtual control and showed that the control actions applied to the pinned nodes can be propagated to the rest of network nodes through the couplings in the network and eventually result in the synchronization of the whole network. In [3], Chen et al. proved that, if the coupling strength is large enough, then even one single pinning controller is able to control network. In the sequel, [4–9] also studied the global pinning controllability of complex networks and some sufficient pinning conditions were established. The common feature of the work in [1–9] is that there are no coupling delays in the network. However, due to the limited speeds of transmission and spreading as well as traffic congestion, signals traveling through a network are often associated with time delays, which are very common in biological and physical networks. Therefore, time delays should be modeled in order to simulate more realistic networks. In [10–13], the pinning synchronization of complex networks with homogeneous time delay is studied. In [14], Xiang et al. considered the pinning control of complex networks with heterogeneous delays via

Abstract:
A class of fuzzy Cohen-Grossberg neural networks with distributed delay and variable coefficients is discussed. It is neither employing coincidence degree theory nor constructing Lyapunov functionals, instead, by applying matrix theory and inequality analysis, some sufficient conditions are obtained to ensure the existence, uniqueness, global attractivity and global exponential stability of the periodic solution for the fuzzy Cohen-Grossberg neural networks. The method is very concise and practical. Moreover, two examples are posed to illustrate the effectiveness of our results.

Abstract:
This paper investigates global exponential synchronization of chaotic systems by designing a novel impulsive controller. The novel impulsive controller is a combination of current and past error states, which is a modification of the normal impulsive one. Some global exponential stability criteria are derived for the error system by utilizing the stability analysis of impulsive differential equations and differential inequalities and, moreover, the exponential convergence rate can be specified. An illustrative example is given to show the effectiveness of the modified impulsive control scheme.