| The stochastic phenomenon widely exists in the fields of biology?finance,communication and control,and is an important factor affecting the nature of the system.When a system is disturbed by random fluctuations,the results become more diverse and complex.Therefore,it is important to study the effects of these stochastic factors on the system's dynamic characteristics.The generation of stochastic factors often makes general differential equations unable to accurately describe the changing laws of system states,so stochastic differential equations are generated.Neutral stochastic differential systems,as a very important type of stochastic delay differential systems,are characterized by not only depicting the derivative terms of the current state,but also considering the derivative terms of the past state.Compared with ordinary stochastic delay differential equations,neutral stochastic delay differential equations can more accurately and profoundly re-flect the law of system changes,and most time-delay systems can be regarded as its special cases.Similarly,stochastic factors are also very important in the dynamics of such systems.Common stochastic factors include time delay,noise,system switching,and so on.As far as noise is concerned,the most commonly used in stochastic differential system is Gauss white noise,which is characterized by continuity and can simulate continuous noise in life,especially in biological neural networks.However,in both natural and engineering problems,noise in-cludes not only continuous noise,but also discontinuous noise.Therefore,Lévy noise can be used to characterize the common effect of two kinds of noise.In addition,due to sudden changes in the external environment or the failure of the system itself,system parameters may change,such as mechanical resonance systems.Such parameter changes can be characterized by the Markov process.Therefore,the research on the stability,synchronization control and optimal control of such neutral stochastic differential systems with noise,time delay and parameter jumps is of far-reaching significance,and related results are currently rare.This thesis focuses on the stochastic differential system,such as stochas-tic neural networks and stochastic neutral differential system,driven by Lévy noise and with Markov switching system parameters.By using Lyapunov stabil-ity theory,It? formula for stochastic differential equation,stochastic inequalities,M-matrix method(or linear matrix inequality),Bellman's optimal principle,etc.,some criteria about adaptive exponential stability and synchronization,cluster synchronization under the Lipschitz condition and non-Lipschitz conditional of stochastic differential system are obtained,respectively.The relative adaptive controllers are designed.The necessary and sufficient condition of realizing opti-mal control for the stochastic differential system is also obtained.The existence condition and express form of Nash equilibrium policy of the non-zero sum dif-ferential game for the stochastic differential system are given.The main researching contents and innovativeness of this thesis are as fol-lows.(1)The exponential synchronization control is considered for neural networks with time-delays and Markovian jumping parameters.The jumping parameters are modeled as continuous-time finite-state Markov chain.By using Lyapunov stability theorem and linear matrix inequality(LMI)technology,the relevant conditions are derived to ensure the global stability of the error system,and the exponential synchronization conditions of the master and slave systems are ob-tained.Numerical simulation is used to verify the feasibility and effectiveness of the proposed synchronization scheme.The innovation is mainly reflected in two aspects:one is to use a generalized time-delay neural network with stochas-tic disturbances and Markov jump parameters as the research model,and its conclusions are more widely used;the other is to consider both discrete time de-lays and distributed time delays for the system state.The obtained criterion of exponential synchronization is an important extension of the results of existing synchronization studies.(2)The adaptive exponential stability of neutral stochastic neural networks with Markov jump parameters on p-th moments is studied.The introduction of Lévy noise makes the neutral neural network more extensive.Combining the generalized It? formula,stochastic analysis,and Lyapunov functional method,an adaptive exponential stability criterion is obtained for a neutral neural network with Markov jump parameters and Lévy noise,and the update law of the adaptive controller and system parameters are given by the analysis method.The validity of the obtained stability criterion is demonstrated by numerical simulation.The innovation is mainly reflected in two aspects:First,the Lévy noise is selected as the external noise of the system.The related results of the stability problem of neutral neural networks with Lévy noise and Markov jump parameters are a powerful supplement to the stability theory.The second is to give a p-th moment exponential stability criterion for generalized neutral neural networks.According to this criterion,a suitable Lyapunov function is chosen to solve the system stability problem caused by the coexistence of neutral terms,Lévy noise and Markov jump parameters.(3)The cluster synchronization problem of coupled neural networks with Lévy noise and Markov transition parameters is studied.Under the event trigger mechanism and the action of the Pinning controller,all nodes in each cluster of the system can achieve synchronization,and the synchronization target node is different between different clusters.The event trigger mechanism can reduce the number of controller updates and control signal transmissions.Based on this,considering the influence of time delay on the system state change and event triggering mechanism,an event triggering Pinning controller related to the current and past state of the system is designed,and corresponding event triggering conditions are given.The Lyapunov stability theory analysis proves the stability of the error system and the cluster synchronization of the coupled neural network.Finally,the effectiveness of the control algorithm is verified by simulation examples.The innovations are mainly reflected in three aspects:First,cluster synchronization based on coupled neural network models is studied.As multi-agents become a research hotspot,cluster synchronization results based on multi-agent models are relatively abundant,but cluster synchronization research on neural network models has just begun.The results in this chapter are a good extension of cluster synchronization research;the second is designing the Pinning controller based on the event trigger mechanism.The update rule depends on the dynamic evolution of the system,and consider the impact of system delay on system state changes and event trigger conditions.This greatly reduces the number of controllers and the number of updates,which helps improve feasibility of practical application and avoid unnecessary energy consumption;Thirdly,the distributed event triggering scheme uses information transmitted by neighboring nodes to determine event triggering conditions,which can effectively exclude Zeno behavior,that is,in any limited time period only trigger a limited number of events.(4)Two kinds of the exponential cluster synchronization of stochastic cou-pled neutral-type neural networks with Lévy noise under non-Lipschitz condition are investigated.Considering the effect of objective time delay on system state changes and event triggering mechanisms,a generalized It? formula and a nonneg-ative semi-marginal convergence theorem are used to design a Pinning controller with corresponding event triggering conditions to derive the stability conditions of the error system.A numerical example presented at the end confirms our theoretical analysis.The innovation is mainly reflected in two aspects:First,the neutral neural network is selected as the system model.Due to the existence of the difference operator of the neutral system,the cluster synchronization cri-terion of the neural network obtained previously can not be directly applied to the neutral neural network.Therefore,it is meaningful to study the problems of mean square exponential cluster synchronization and almost asymptotic ex-ponential cluster synchronization of neutral coupled stochastic neural networks.Second,the non-Lipschitz condition is more general than the Lipschitz condition mentioned in Chapter 4.The range of activation functions is also wider.(5)A neutral non-zero and linear quadratic stochastic differential game model is establish.The model contains neutral terms related to the current and past states,and also reflects the rate of change of the state and the effects of time delay and noise on the state of the system.The definition of neutral differential game is given for the first time,and two different game strategies are given under non-zero sum.These two are linear feedback strategies,and the difference is whether to consider the influence of past states on system states and strategies.This game problem can be equivalent to the existence of the solution of a fourth-order coupled stochastic Riccati equation under the assumption of the mean square stability of a stochastic system.By solving this equation,the condi-tions for the existence of the game Nash strategy are obtained.To illustrate the practicality of the results,two practical examples are given.The first example proves that the results can effectively solve the stochastic H2/H? control problem in an infinite time range.The second financial example illustrates in detail how the model works.The innovation points are mainly reflected in three aspects:The first is to introduce the concept of neutral terms into differential games.The use of neutral stochastic differential games to derive Nash equilibrium strategies will be more general than the conclusions of existing differential games;The sec-ond is to consider the influence of past states on the choice of game strategy,which is more in line with the laws of the real world;The third is to apply the conclusions of the neutral stochastic differential game to the stochastic H2/H?control problem and financial investment strategy issues. |
PDF Full Text Request