Stability Analysis And Optimal Control Of Several Stochastic Systems Driven By Lévy Process

Up to now, the noise of the stochastic differential system is generally Gauss process whose characteristic is continuity. However, the stochastic system is disturbed not only by Gauss continuous noise, but also by Possion noncontinuous noise. For example, the interference of the global financial crisis triggered by the stock market sharply oscillation is discontinuous. The combination of the two noise of the stochastic system can be characterised by Lévy noise. Moreover, in real engineering, there exist many systems, such as mechanical oscillation system,whose parameters are jump. This kind of jump parameters can be characterised by Markov process. Therefore, in the real systems, there widely exist the stochastic differential systems driven by Lévy noise and with Markov switching parameters. However, the study of stability and optimal control for this kind of systems is rare yet.This thesis focuses on the stochastic differential system, such as stochastic neural networks and stochastic financial system, driven by Lévy noise and with Markov switching system parameters. By using Lyapunov stability theory,It?o formula for stochastic differential equation, stochastic inequalities, M-matrix method(or linear matrix inequality), Bellman's optimal principle, etc., some criteria about adaptive almost surely asymptotic stability, adaptive stability in mean square, and adaptive exponential stability of stochastic differential system are obtained, respectively. The relative adaptive controllers are designed. The portfolio policy of financial market is also analysed and designed which has been applied into European option pricing problem. The necessary and sufficient condition of realizing optimal control for the stochastic differential system is also obtained. The existence condition and express form of Nash equilibrium policy of the non-zero sum differential game for the stochastic differential system are given.The above research of this thesis can be used to solve the key application problem of steady output, coincident response and control efficiency for the s-tochastic differential system driven by Lévy noise, and can enrich the stability and optimal control theory for the stochastic differential systems.The main researching contents and innovativeness of this thesis are as follows.(1) Study the adaptive asymptotic stability for the neutral-type neural network with stochastic disturbance and Markov switching parameters. After introducing and analyzing the existing study, we set up the neutral-type neural networks model with Gauss noise and Markov switching parameters. Under some necessary assumptions, we obtain the almost surely asymptotic stability criteria and the adaptive control law of the system by using M-matrix method and Lyapunov functional technique. Some numerical simulation examples are developed to illustrate the effective of the controller and to verify the stability of the system. The novelty is adding an adjust term in designing controller such that the control effective well. Furthermore, the stability criteria obtained by using M-matrix method are simple and easy to judge.(2) Consider the analysis problem of synchronization in mean square for neural network with Lévy noise and Markov switching parameters based on data sampling and saturating actuator. Firstly, after analysing the existing study,we point its disadvantages, and introducing our study contents and contribution.Secondly, we set up neural network model with Lévy noise, Markov switching parameters and multiple slaver system. Some necessary assumptions and lemmas are given. In setting up model, the data sampling and saturating actuator are introduced. Thirdly, the criteria of synchronization in mean square of the system and the gain matrix of the controller are obtained by using weakly infinite operator, Dynkin formula, and linear matrix inequality method. We then give two corollaries for two special models. Finally, some numerical simulation examples are developed to illustrate the effective of the method and results obtained. The novelty is the system model which bases on data sampling and saturating actuator, and the synchronization condition in mean square obtained by considering the data sampling and saturating actuator.(3) Consider adaptive exponential synchronization problem of one master and multiple slaver neural networks with Lévy noise and Markov switching parameters. In our model, the parameter of the slaver system is estimated, and thestate matrix of the slaver system is relative not only the master system and also the other slaver systems. We design the controller by making using of the errors of the system. To speed the synchronization, we design exponential synchronization control law. By using general It?o formula, Lyapumov stability theorem and M-matrix method, some adaptive synchronization criteria are given such that the slaver system is exponential synchronization with master system. Meanwhile, we obtain the update law of the controller and dynamic equation of the slaver system parameters. Finally, a numeral example is given to verify the effective of the control method. The novelty is proposing one master and multiple slaver neural network model and giving a new adaptive synchronization control method.(4) Consider portfolio problem of financial market with Lévy noise and Markov switching regime. Firstly, a new portfolio model of financial market with Lévy noise and Markov switching regime is set up. This model is an extension of Black-Scholes model in financial market. In the new model, interest rate of the bond, the rate of return and the volatility of the stocks vary as the market states switching and the stock prices are driven by geometric Lévy process. For the new model of the financial market, a portfolio strategy which is determined by a partial differential equation(PDE) of parabolic type is given by using It?o formula. The PDE is an extension of existing result. The solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally. The novelty is setting up the new portfolio model of financial market with Lévy noise and Markov switching and obtaining the portfolio policy by using of It?o's formula.(5) Study on optimal control and non-zero num game problem for the stochastic differential system with Lévy noise and Markov switching parameters.Firstly, we set up the stochastic differential system model with Lévy noise and Markov switching, and give the concept of Nash equilibrium point. Secondly,we obtain Hamiltonian-Jacobi-Bellman(HJB) equation and the solution of the inverse problem by using of Bellman dynamic program. Based on the HJB equation, the solution of linear quadratic Gauss optimal control problem is given. For the linear quadratic Gauss differential game problem, the solution of the nonzero sum game problem is obtained by using of the solution of optimal controlproblem. Finally, we obtain the existing condition of Nash equilibrium point by using the results in the game problem of stock market. The novelty is that the optimal control law and the Nash equilibrium point are obtained by using of Bellman dynamic program principle for the stochastic differential system with Lévy noise and Markov switching.