Exponential Stability Analysis And Synchronization Control Of Stochastic Systems With Markovian Switching Parameters

Due to the change of coupling between subsystems,sudden environmental interference,the parameters and structure of the system may change abruptly in practical applications such as the economic system,aircraft control system,and robot operating system.In order to describe the system more appropriately,hybrid dynamic systems with both discrete and continuously variable operating mechanisms are generally used for modeling.The transformation between modes can be expressed in many forms.Among them,due to the particular Markovian property of Markovian chains,differential equations with Markovian switching are usually used to solve this problem.In the past few decades,the main achievements in this field have been applied to stability analysis,filtering,optimization,and some important control problems.Most applications of neural networks require increasing the convergence speed of the network for shortening the calculation time of neural networks.When the exponential convergence speed is used to determine the calculation speed of the neural networks,the exponential stability characteristic is even more important.Therefore,it is not only interesting in theory but also valuable in practice to determine the exponential stability of dynamic systems and estimate their exponential convergence rate.Besides,since the phenomenon of synchronization has been observed,more and more researchers have paid attention to synchronization issues,including generalized synchronization and cluster synchronization.At present,there are still many gaps in exponential stability analysis and synchronization control of Markovian switched stochastic systems.Therefore,it is still an interesting and challenging work to study the exponential stability analysis and synchronization control of stochastic systems with Markovian switching parameters.This thesis focuses on the stochastic systems with time-varying delays and Markovian switching parameters,Lyapunov stability theory,stochastic analysis method,linear matrix inequality method and random inequality are used to obtain the mean-square exponential stability conditions and the almost surelyexponential stability conditions;Combined with feedback control,sliding mode control and pinning control,the corresponding synchronization controllers are designed.The structure-triggered asymptotical synchronization conditions of nonidentical generalized stochastic systems with Markovian jumping parameter and the finite-time and fixed-time cluster synchronization criteria of the delayed neural networks are given respectively.The above research work will enrich the stability and synchronization theory of stochastic systems,and solve the key problems of the exponential stability of Markovian switching neutral stochastic systems with general transition probability and general noise.The main researching contents and innovativeness of this thesis are as follows.(1)Study on the problem of exponential stability for Markovian switching neutral stochastic systems with general transition probabilities and time-varying delay.Based on non-convolution type multiple Lyapunov functions and stochastic analysis method,we obtain the mean-square exponential stability conditions and the almost surely exponential stability conditions of Markovian neutral stochastic systems with general transition probability and time-varying delay.Two numerical simulation examples are presented to illustrate the effectiveness and potential of the proposed results.The novelty is the conditions which are independent to any decay rate of the exponential stability for uncertain transition probabilities neutral stochastic systems with time-varying delay.(2)Consider the problem of exponential stability of neural networks with Markovian parameters and general noise.Criteria for the exponential stability of the neural networks with Markovian switching parameters and general noise in both the mean square and p-th moment are derived by utilizing the random analysis method and Lyapunov functional method techniques.The exponential stability of neural networks without Markovian switching is given as a special case.Simulation result in two examples are discussed to illustrate the theoretical results.The novelty is the model in this paper with general noise is more suitable for many real nervous systems than neural networks with white noise.(3)Consider the problem of structure-triggered asymptotical synchronization for nonidentical generalized stochastic systems with Markovian jumping parameter.By introducing a structure-triggered communication scheme,the opti-mal master system is chosen for the slave system to synchronize.Furthermore,if the master system selected for the first time fails,the system will be promptly chosen again.Based on the Lyapunov stability theory,sliding mode control approach and the linear matrix inequality technique,some sufficient conditions for synchronization of the nonidentical generalized stochastic systems are obtained.The applicability of the method is illustrated by two numerical examples.The novelty is the stochastic system model with structure-triggered communication scheme and the design of the stochastic sliding surfaces for the error systems of nonidentical generalized Markovian stochastic systems.(4)Study on the problems of finite-time and fixed-time cluster synchronization of the delayed neural networks.Based on the Lyapunov stability theory,the finite time and fixed time synchronization criteria are derived.In addition,the settling time for stabilization that is dependent on initial value and independent of the initial value is estimated respectively.In order to realize cluster synchronization with finite time and fixed time,some simple distributed protocols with pinning control are designed and proved to be effective.A numerical simulation is used to illustrate the correctness and effectiveness of the proposed methods.The novelty is the neural networks considered the coupling relationship of nodes in the same cluster and nodes in different clusters which is more complicated.