Stability Of Differently Structured Hybrid Highly Nonlinear Neutral Stochastic Delay Differential Equations

As an important field of stochastic analysis,stochastic differential equations play a great role in many areas of applied and theoretical value.This thesis aims at studying the stability of highly nonlinear hybrid neutral stochastic delay differential equations(NSDDEs)with different structures.The main characteristics are as follows: the new system is nonlinear,that is,the coefficients do not satisfy the linear growth condition;we consider the factors including the neutral term,time delay term and the Markov switching in our system at the same time;the system has completely different system structures in different switching state spaces of Markov chain.There are three main works in this thesis: the exponential stability of highly nonlinear hybrid NSDDEs with multiple time-varying delays and different structures,the exponential stability of highly nonlinear hybrid NSDDEs with unbounded delays and different structures,and the asymptotic stabilization by delay feedback control for highly nonlinear hybrid NSDDEs with different structures.In terms of the research objects,the diversity and complexity of real dynamic systems impel us to investigate various types of stochastic differential equations necessarily.One class of them is the NSDDE,which is usually used to characterize the stochastic systems depending on both present and past states with its changing rate.For example,the impact of economic policies on the stock market often has a certain time delay,and as we all know,the COVID-19 has an incubation period.Another one is the hybrid NSDDE with Markovian switching,which can be used to describe the situation that above stochastic systems encounter abrupt changes due to the Internal or external environments,such as the bull and bear markets in the stock market,the state switching of air traffic or power systems,etc.While in most of the existing literatures,the hybrid NSDDEs have exactly the same structure in each subspace but different coefficients.Therefore,this thesis aims to study the stability of highly nonlinear hybrid NSDDEs with different structures.In terms of the research content,the research of stochastic differential equations mainly focuses on the existence and uniqueness of the solution,the approximation of numerical solution and the stability.In classical results,the Lipschitz condition and linear growth condition are needed to ensure the existence and uniqueness of the solution.However,these two conditions are too strict to be satisfied in many practical cases.In recent years,with the development of scientific research,much more attentions are paid to the highly nonlinear stochastic systems with the weaker local Lipschitz condition.On the other hand,the study of stability of stochastic differential equations mainly includes the pth moment asymptotic stability,the almost surely asymptotic stability,the pth moment exponential stability and almost surely exponential stability.Besides,there are also many methods on the numerical solution approximation of stochastic differential equations,such as the Euler-Maruyama method and the Caratheodory method.In this thesis,we mainly use the Lyapunov function method and M-matrix method to study the exponential stability and asymptotic stability of highly nonlinear hybrid neutral delay stochastic differential equations with different structures,as well as the numerical solution approximation based on the Euler-Maruyama method.There are three key works in this thesis: Firstly,we study the exponential stability of highly nonlinear hybrid NSDDEs with multiple time-varying delays and different structures.Based on the differently structured stochastic switching situation,we generalize the usual single constant delay to multiple function delays in the new system,and the delays contained in the coefficients are also different.The existence,uniqueness and asymptotic boundedness of the solution are proved under local Lipschitz condition and nonlinear growth conditions.Then the criteria of the pth moment exponential stability and almost surely exponential stability are given.Besides,the Euler-Maruyama numerical solution is defined.We also prove that it converges to the theoretical solution in probability.And a specific numerical example is given to illustrate the main results.Secondly,we study the exponential stability of highly nonlinear hybrid NSDDEs with unbounded delays and different structures.Based on the differently structured stochastic switching situation,we extend the above conclusions to the case of unbounded time delays,which enriches the relevant theoretical results of such equations.The extending of the delay terms in the system from the boundedness to unboundedness makes our results more general and applicable,but also improves the difficulty of the theoretical analysis.Finally,we study the asymptotic stabilization by delay feedback control for highly nonlinear hybrid NSDDEs with different structures.When the given highly nonlinear differently structured hybrid NSDDE is unstable,we design a delay feedback control function satisfying the Lipschitz condition in the system to make the controlled system asymptotically stable.The Lyapunov function method,M-matrix method and the generalized It(?) formula are mainly used to investigate the design criteria of the delay feedback controller.And then we establish the asymptotically stable criteria of the new system.In summary,stochastic differential equations and stochastic control systems are quite important and have been widely applied in the fields of finance,biology,physics,engineering and so on.Based on the situation that neutral stochastic systems have completely different structures in different Markovian switching subspace,we study the exponential stability and asymptotic stability of highly nonlinear hybrid NSDDEs,and obtain more general results.This kind of model is of much research value and the results obtained enrich the theoretical basis for the applications of stochastic differential equations.