M-Matrix Strategies For Stability Analysis And Synchronization Control Of Impulsive Delayed Neural Networks

After continuous development and improvement, neural networks has been successfully applied to many fields such as computer science, artificial intelligence, automatic control, pattern recognition, image processing, combinatorial optimization, associative memory, and so on. Such applications deeply rely on the dynamical behaviors of neural networks. In practice, noting that delays exist objectively and impulses occur suddenly,and it has been proved that they have an significant influence on the dynamical behaviors of neural networks. So, the investigation on dynamics of neural networks with delays and impulses, which attracts many scholars’ attentions, has been one of the important research on dynamics of non-linear system.Based on the predecessors’ work, the dynamics of several class of neural networks with delays of different type have been thoroughly investigated under impulsive effects.Generally speaking, it includes that the problems of stability related to the equilibrium point and the periodic orbit under impulsive perturbations, the problem of stabilization related to the equilibrium point via impulsive control, the problem of chaotic synchronization under impulsive perturbations and impulsive control, respectively. In details, the main contents are summarized in the following six aspects:(I) The problem of stability related to the equilibrium point is investigated for a general neural networks with continuously distributed delays and nonlinear impulses, which incorporated impulsive delayed Hopfield neural networks and cellular neural networks as its special case. Firstly, by employing topological degree, M-matrix and inequalities technique, a criterion is obtained in terms of M-matrix, which ensures the existence and uniqueness of equilibria of the considered neural networks model. Subsequently, it is proved that the M-matrix condition for the existence and uniqueness of the equilibrium point also implies its global exponential stability under some proper nonlinear impulsive perturbations by applying the analysis method. Finally, an illustrative example is given to show the validation and superiority of the theoretical results.(II) The problem of stability related to the equilibrium point is further investigated for the neural networks considered in I under the variable delays and the nonlinear impulses of general type, which can be viewed as the continuous work in I. Different from the topological degree method used in I, based on the homeomorphism theory, M-matrix and inequalities technique, a criterion is obtained in general terms of M-matrix, which ensures the existence and uniqueness of equilibria of the considered neural networks model.After that, it is proved that the M-matrix condition for the existence and uniqueness of the equilibrium point also implies its global exponential stability under some general nonlinear impulsive perturbations by applying the analysis approach. Finally, two illustrative examples are given to show the validation and superiority of the theoretical results.(III) The problem of stability related to the equilibrium point is investigated for a class of static neural networks with proportional delays and linear impulses, where the neuron states are utilized as basic variables. The proportional delays considered herein is a kind of unbounded time-varying delay, which is different from the continuously distributed delays in I and the bounded variable delays in II. Based on the generalized matrix measure and Halanay inequality, criteria are obtained in terms of matrix measure, which guarantee the global exponential stability of equilibria of the considered neural networks model. Finally, two illustrative examples are given to show the validation and superiority of the theoretical results.(IV) The periodic oscillation is investigated for a type of high-order BAM neural networks with periodically variable coefficients and external inputs, continuously distributed delays and nonlinear impulses. Based on M-matrix, Lyapunov-Krasovskii functional method, inequality and analysis technique, certain sufficient conditions are obtained to ensure the existence, uniqueness and global exponential stability of the periodic solution of the considered neural networks model. Finally, an illustrative example is given to show the validation and superiority of the theoretical results.(V) The problem of impulsive stabilization is investigated for a kind of memristive neural networks(By replacing the resistors with memristors in traditional neural networks, memristive neural networks can be obtained at once). From the impulsive control point of view, algebraic inequality criteria guaranteeing the global exponential stability of equilibria of the considered neural networks model have been presented by utilizing set-valued mapping, impulsive differential inclusions, non-smooth analysis and establishing a new impulsive differential inequality. Finally, two illustrative examples are given to check the effectiveness and feasibility of the theoretical results.(VI) Based on two different impulsive effects(impulsive perturbations and impulsive control), the problem of synchronization control is investigated for two kinds of chaotic memristive neural networks by means of drive-response concept, where the first type is that the drive system is subjected to impulsive disturbances and the other type is that only the response system is added by useful impulse control. For these two situations,state feedback and impulsive control synchronization criteria are obtained by exploiting Halanay inequality of impulsive perturbation type, Halanay inequality of impulsive control type and the ideas in II and V, respectively. Finally, an illustrative example is given to check the effectiveness and feasibility of these two synchronization criteria.