Abstract:
The global exponential robust stability is investigated to a class of reaction-diffusion Cohen-Grossberg neural network (CGNNs) with constant time-delays, this neural network contains time invariant uncertain parameters whose values are unknown but bounded in given compact sets. By employing the Lyapunov-functional method, several new sufficient conditions are obtained to ensure the global exponential robust stability of equilibrium point for the reaction diffusion CGNN with delays. These sufficient conditions depend on the reaction-diffusion terms, which is a preeminent feature that distinguishes the present research from the previous research on delayed neural networks with reaction-diffusion. Two examples are given to show the effectiveness of the obtained results.

Abstract:
The problems of delay-dependent exponential passivity analysis and exponential passification of uncertain Markovian jump systems (MJSs) with partially known transition rates are investigated. In the deterministic model, the time-varying delay is in a given range and the uncertainties are assumed to be norm bounded. With constructing appropriate Lyapunov-Krasovskii functional (LKF) combining with Jensen’s inequality and the free-weighting matrix method, delay-dependent exponential passification conditions are obtained in terms of linear matrix inequalities (LMI). Based on the condition, desired state-feedback controllers are designed, which guarantee that the closed-loop MJS is exponentially passive. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Abstract:
Stochastic effects on convergence dynamics of reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with delays are studied. By utilizing Poincaré inequality, constructing suitable Lyapunov functionals, and employing the method of stochastic analysis and nonnegative semimartingale convergence theorem, some sufficient conditions ensuring almost sure exponential stability and mean square exponential stability are derived. Diffusion term has played an important role in the sufficient conditions, which is a preeminent feature that distinguishes the present research from the previous. Two numerical examples and comparison are given to illustrate our results.

Abstract:
The mean square BIBO stability is investigated for stochastic control systems with mixed delays and nonlinear perturbations. The system with mixed delays is transformed, then a class of suitable Lyapunov functionals is selected, and some novel delay-dependent BIBO stabilization in mean square criteria for stochastic control systems with mixed delays and nonlinear perturbations are obtained by applying the technique of analyzing controller and the method of existing a positive definite solution to an auxiliary algebraic Riccati matrix equation. A numerical example is given to illustrate the validity of the main results.

Abstract:
Stochastic effects on convergence dynamics of reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with delays are studied. By utilizing Poincaré inequality, constructing suitable Lyapunov functionals, and employing the method of stochastic analysis and nonnegative semimartingale convergence theorem, some sufficient conditions ensuring almost sure exponential stability and mean square exponential stability are derived. Diffusion term has played an important role in the sufficient conditions, which is a preeminent feature that distinguishes the present research from the previous. Two numerical examples and comparison are given to illustrate our results.

Abstract:
We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible. 1. Introduction Graph theory has developed into a substantial body of knowledge. A graph theoretic approach developed in [1, 2] is used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multigroup models in mathematical epidemiology. Using results from graph theory, a systematic approach developed in [3] allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. The graph-theoretical approach has been applied to various classes of coupled systems in engineering, ecology, and epidemiology (see, e.g., [1–13]). However, as in [14], while using the same Lyapunov function [3], sometimes graph theory can be replaced by positive operator theory. Furthermore, it seems that all authors use the graph theory under the assumption that the relative matrix is irreducible (see, e.g., [1–13]). Motivated by the above discussion, in this paper, we show some results which can replace graph theory used to construct global Lyapunov functions in some coupled systems of differential equations (see, e.g., [1–13]) and a more general threshold without the assumption that the relative matrix is irreducible. For some coupled systems of differential equations (see, e.g., [1–13]), by a similar approach to the one discussed in this paper, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible. Various epidemics continue to pose a public health threat to humans. One of the most important subjects in this study of epidemic models (see, e.g., [1–3, 7–13]) is to obtain a threshold that determines the persistence or extinction of a disease. In the real world, some epidemics, such as malaria, dengue, fever, gonorrhea, and

Abstract:
This paper concerns the problem of delay-dependent stability criteria for recurrent neural networks with time varying delays. By taking more information of states and activation functions as augmented vectors, a new class of the Lyapunov functional is proposed. Then, some less conservative stability criteria are obtained in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Abstract:
This paper address the problems of robust stability for uncertain discrete-time switched systems. The uncertainty is assumed to be of structured linear fractional from which includes the norm-bounded uncertainty as a special case. By introducing a novel dierence inequality, new delay-dependent stability criteria are formulated in terms of linear matrix inequalities(LMIs) which are not contained in known literature. Numerical examples are given to demonstrate the eectiveness of the theoretical results.

Abstract:
The problem of bounded-input bounded-output (BIBO) stabilization for discrete-time uncertain system with time delay is investigated. By constructing an augmented Lyapunov function, some sucient conditions guaranteeing BIBO stabilization and robust BIBO stabilization are established. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Two numerical examples are provided to demonstrate the eectiveness of the derived results.