Research On Multistability And Hybrid Control For Several Classes Of Complex Nonlinear Systems

Examples of complex nonlinear systems with the features such as random-ness, time-delays, impulses, switching and reaction-diffusion can be found in the nature and practical engineering applications. Influenced by nonlinear evolution of state variables, hybrid driving governed by interacting discrete and continu-ous dynamics, and the uncertain effects of random factors, such systems exhibit complex dynamic behaviors. The research topics for analysis and control of com-plex nonlinear systems include developing effective mathematical methods for analyzing their dynamical properties, and designing practical and efficient con-trol strategies by exploring their structure characteristics and evolution laws. This thesis is focused on analysis and control problems of several classes of complex nonlinear systems. By fully exploiting their structure features, the multistability properties and periodic phenomena of these systems are studied. By analyzing the driving mechanisms of three types of hybrid control methods, including im-pulsive control, sampled-data control, and intermittent control, several efficient hybrid control strategies for these systems are developed.(1) Multistability analysis for delayed stochastic Hopfield neural networks is investigated. By considering the geometrical configuration of activation func-tions, the state space is split into 2n+1 regions which contain 2n unbounded rectangular regions. By applying Schauder’s fixed-point theorem and some nov- el stochastic analysis techniques, it is shown that under some conditions, the 2n rectangular regions are positively invariant with probability one, and each of them possesses a unique equilibrium. Then with respect to the slowly-varying delay and fast-varying delay, by applying Lyapunov function approach and functional approach, two multistability criteria are established for ensuring these equilibria to be locally exponentially stable in mean square.(2) The existence of multiple periodical solutions, stability and anti-disturbance problem are investigated for delayed stochastic cell neural networks. By using the contraction mapping theory, some novel stochastic analysis techniques and Lyapunov functional, a sufficient condition of the existence and uniqueness for stochastic delayed neural networks is established. Then, the stochastic periodici-ty with disturbance attenuation of delayed cell neural networks is further studied. Our results suggest that multistability of periodical solutions have certain robust-ness to stochastic perturbation.(3) The problem of generating of periodic orbit and control for delayed neu-ral networks via impulsive control is considered. By choosing a "weighted" phase space PCα, establish a general criterion for the existence, uniqueness, and glob-ally exponential stability of periodic solution of neural networks with unbounded distributed delay. Then, based on weighted impulse-time-dependent Lyapunov function/functional analysis approaches, some control strategies for generating globally exponentially stable periodic solutions under periodic impulsive are es-tablished.(4) The impulsive synchronization problem of reaction-diffusion neural net-works with mixed delays and its application to image-encrypting cryptosystem is studied. First, the new criterion is established by applying an impulse-time-dependent Lyapunov functional technique, and applying improved Wirtinger in- equality to deal with the reaction-diffusion terms. Comparing with previous re-sults, the new criterion can be largely reduces the conservatism. Then by applying the obtained synchronized results, build an image-encrypting decryption algorith-m based on spatiotemporal chaotic impulsive synchronization, and also construct a secret communication system which can transmit an encrypted image. Final-ly, security analysis which includes key space analysis, key sensitivity analysis, statistical analysis and information entropy analysis of encryption algorithm is discussed through an experimental simulation. The experimental results verify that the proposed image-encrypting cryptosystem has the advantages of large key space and high security against some traditional attacks.(5) The finite dimensional intermittent stabilization problem of reaction-diffusion neural networks with point measured output is studied. The proposed stabilization scheme is based on the assumption that the state is point sampled in space and the control action is intermittently activated in time. That is, the sampled-data control is only activated in "work time", and at each moment dur-ing "work time", the measurement of the states is taken in a finite number of fixed sampling points in the spatial domain. By introducing a piecewise Lyapunov function combined with the technique of applying Wirtinger’s inequality for ex-ploring the stabilizing role of the reaction-diffusion term, sufficient conditions for globally mean-square exponential stability is developed in terms of linear matrix inequalities. The obtained conditions establish a quantitative relation among the bound of control time intervals, rest time intervals and the step-size of spatial sampling. Based on these stability criteria, some parameterized representation of the periodically intermittent sampled-data controller is presented.(6) The sampled-data distributed H∞ control problem for one-dimensional semilinear transport reaction equations with external disturbances is considered.Based on sampled information of state on a series discrete time of infinite points in spatial, a Razumikhin-Lyapunov functional analysis technique is proposed. Then some criteria for internal exponential stability and finite L2-gain are derived in terms of linear matrix inequalities. The obtained criteria establish a quantitative relation among the step-size of spatial sampling and the step-size of time sam-pling, and L2-gain. In contrast to Halanay’s inequality based approach which was proposed by Fridman, the proposed Razumikhin-type approach not only per-fectly solves the problem of sampled-data distributed H∞, but also reduces the conservative of Fridman’s results about stabilization of sampled-data distributed systems.