Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Impulsive stabilization of stochastic functional differential equations

AbstractThis paper investigates impulsive stabilization of stochastic delay differential equations. Both moment and almost sure exponential stability criteria are established using the Lyapunov–Razumikhin method. It is shown that an unstable stochastic delay system can be successfully stabilized by impulses. The results can be easily applied to stochastic systems with arbitrarily large delays. An example with its numerical simulation is presented to illustrate the main results

Nonlinear analysis of dynamical complex networks

Copyright © 2013 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Complex networks are composed of a large number of highly interconnected dynamical units and therefore exhibit very complicated dynamics. Examples of such complex networks include the Internet, that is, a network of routers or domains, the World Wide Web (WWW), that is, a network of websites, the brain, that is, a network of neurons, and an organization, that is, a network of people. Since the introduction of the small-world network principle, a great deal of research has been focused on the dependence of the asymptotic behavior of interconnected oscillatory agents on the structural properties of complex networks. It has been found out that the general structure of the interaction network may play a crucial role in the emergence of synchronization phenomena in various fields such as physics, technology, and the life sciences

Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems

According to Newton's second law of motion, we humans describe a dynamical system with a differential equation, which is naturally discretized into a difference equation whenever a computer is used. The differential equation is the physical model in human brains and the difference equation the cyber model in computers for the dynamical system. The physical model refers to the dynamical system itself (particularly, a human-designed system) in the physical world and the cyber model symbolises it in the cyber counterpart. This paper formulates a hybrid model with impulsive differential equations for the dynamical system, which integrates its physical model in real world/human brains and its cyber counterpart in computers. The presented results establish a theoretic foundation for the scientific study of control and communication in the animal/human and the machine (Norbert Wiener) in the era of rise of the machines as well as a systems science for cyber-physical systems (CPS)

Stability of hybrid stochastic retarded systems

Abstract-In the past few years, hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), including hybrid stochastic delay systems, have been intensively studied. Among the key results, Mao et al. proposed the Razumikhin-type theorem on exponential stability of stochastic functional differential equations with Markovian switching and its application to hybrid stochastic delay interval systems. However, the importance of general asymptotic stability has not been considered. This paper is to study Razumikhin-type theorems on general theorem moment asymptotic stability of hybrid stochastic retarded systems. The proposed theorems apply to complex systems including some cases when the existing results cannot be used

Robust normalization and guaranteed cost control for a class of uncertain singular Markovian jump systems via hybrid impulsive control

This paper investigates the problem of robust normalization and guaranteed cost control for a class of uncertain singular Markovian jump systems. The uncertainties exhibit in both system matrices and transition rate matrix of the Markovian chain. A new impulsive and proportional-derivative control strategy is presented, where the derivative gain is to make the closed-loop system of the singular plant to be a normal one, and the impulsive control part is to make the value of the Lyapunov function does not increase at each time instant of the Markovian switching. A linearization approach via congruence transformations is proposed to solve the controller design problem. The cost function is minimized via solving an optimization problem under the designed control scheme. Finally, three examples (two numerical examples and an RC pulse divider circuit example) are provided to illustrate the effectiveness and applicability of the proposed methods

Practical exponential stability of impulsive stochastic functional differential equations

This paper is devoted to the investigation of the practical exponential stability of impulsive stochastic functional differential equations. The main tool used to prove the results is the Lyapunov-Razumikhin method which has proven very useful in dealing with stability problems for differential systems when the delays involved in the equations are not differentiable but only continuous. An illustrative example is also analyzed to show the applicability and interest of the main results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí