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This paper is concerned with the global stability of the nonlinear model for genetic regulator networks (GRNs) with time delays. Four new sufficient and necessary conditions for global asymptotic stability and global exponential stability of the equilibrium point of GRNs are derived. Specifically, using comparing theorem and Dini derivation method, three weak sufficient conditions for global stability of GRNs with constant time delays are proposed. Finally, a general GRN model is used to illustrate the effectiveness of the proposed theoretical results. Compared with the previous results, some sufficient and necessary conditions for Lyapunov stability of GRNs are proposed, which are not seen before. 本文讨论了一类非线性时滞基因调控网络的全局稳定性,给出了四个关于基因调控网络平衡点全局渐近稳定和全局指数稳定的充分必要条件。此外,本文通过运用比较定理和Dini导数,提出了三个关于基因调控网络全局稳定的充分条件。最后,通过一个典型的基因调控网络模型,进一步验证了本文给出结论的有效性。与已有基因调控网络稳定性的结论相比,基于Lyapunov稳定性的充分必要条件在以前的文献中未见到。

Abstract: The sufficient and necessary conditions for Lyapunov stability of the zero equilibrium point of Lorenz system are discussed, and some brief criteria are presented for globally exponential stability, globally asymptotical stability and instability. Furthermore, the behavior of the non-zero equilibrium point of Lorenz system is also investigated, and several sufficient and necessary conditions are provided for locally exponential stability and instability. The established theorems in this paper develop and extend the existing achievements on Lyapunov stability of Lorenz system. In conclusion, by applying the obtained results, some less conservative feedback-control laws are derived to guarantee the globally exponential stability of the chaos control of Chen system, Lü system, Yang-Chen system and Yu-Xia Li system.

Abstract: This paper presents some simple feedback control laws to study global stabilization and global synchronization for a special chaotic system described in the generalized Lorenz canonical form (GLCF) when τ =−1 (which, for convenience, we call Shimizu–Morioka system, or simply SM system). For an arbitrarily given equilibrium point, a simple feedback controller is designed to globally, exponentially stabilize the system, and reach globally exponent synchronization for two such systems. Based on the system’s coefficients and the structure of the system, simple feedback control laws and corresponding Lyapunov functions are constructed. Because all conditions are obtained explicitly in terms of algebraic expressions, they are easy to be implemented and applied to real problems. Numerical simulation results are presented to verify the theoretical predictions. [Copyright &y& Elsevier]

Abstract: This paper presents a new method for controlling chaos in several classical chaotic Rössler systems using feedback control strategy. In particular, for an arbitrarily given equilibrium point of a Rössler system, we design explicit and simple feedback control laws by which the equilibrium point is globally and exponentially stabilized. Six typical Rössler systems are studied, and explicit formulas are derived for estimating the convergence rate of these systems. Numerical examples are presented to illustrate the theoretical results. A mistake has been found in the existing literature, and a correct result is given. [Copyright &y& Elsevier]

We discuss the robust dissipativity with respect to the quadratic supply rate for uncertain impulsive dynamical systems. By employing the Hamilton-Jacobi inequality approach, some sufficient conditions of robust dissipativity for this kind of system are established. Finally, we specialize the obtained results to the case of uncertain linear impulsive dynamical systems.

In this paper, we transform n–th order Lure direct control systems into nonlinear systems with separating variables, and induce a concept for absolute stability of part arguments. The necessary and sufficient conditions (NASC) for the absolute stability of Lure direct control systems are given and some sufficient conditions are obtained

Abstract: This paper is concerned with the exponential stability for functional differential equations. A generalized Halanay inequality is established. By applying the generalized Halanay inequality and Lyapunov functional method, new sufficient conditions are obtained ensuring the global exponential stability of the trivial solution of the equations. Two examples are also given for illustration. [Copyright &y& Elsevier]