Stability Of Complex Nonlinear Systems

According to the properties of M-matrix theory and the technique of vector Lyapunov function, by applying differential inequalities, the stability of four classes of nonlinear systems is analyzed.1. The existence and uniqueness of the equilibrium point, and absolute exponential stability for a class of generalized Hopfield neural networks (GHNN) are investigated. The Hopfield neural networks and cellular neural networks are the special cases of the GHNN. The activation functions of the GHNN are not necessary to be a differential and bounded function. By applying topology theory, the necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the GHNN is obtained. By constructing Lyapunov functions in Lurie type, and using the matrix property, the sufficient conditions of absolute exponential stability for GHNN are given.2. The global exponential stability of a class of interconnected nonlinear large-scale systems with time delays is studied. A criterion is given for global exponential stability of the systems by analyzing the stability of differential inequalities with time delays under the assumption that the time delays are bounded and continuous.3. The global exponential stability of a class of nonlinear interconnected interval large-scale systems with time delays is analyzed. A criterion is obtained for global exponential stability of the systems by analyzing the stability of differential inequalities with time delays under the assumption that interconnected part is globally Lipschitz. Some sufficient conditions are given for global exponential stability of interval neural networks systems with time delays. Since criterions obtained are independent of the time delays, they are easy for application.4. Asymptotic stability of interconnected system and singularly perturbed interconnected system is introduced. Some sufficient conditions are obtained for asymptotic stability of interconnected systems. By dividing singularly perturbed interconnected system into two subsystems which are the reduced system and the boundary system, a criterion is studied for asymptotic stability of singularly perturbed interconnected system. A new technique is provided for the design of nonlinear singularly perturbed system.