First Hitting Time Problems For Uncertain Fractional-order Dynamical Systems

First hitting time is a kind of indeterminate time considered firstly in history,and has been widely applied in queuing theory,bankruptcy problem,as well as the reliability,maintenance and quality control problems.Also,first hitting criteria is a criterion of a system to achieve some performance index before the target state set for the first time,and taking the first hitting criteria as the objective function,such optimal control problem is also a significant extension of the optimal control theory.Corresponding time optimal control problem aims at seeking the optimal decision in a dynamic system to optimize the time of decision makers.In addition,the actual dynamic system can be interfered by different kinds of noise when it is running,and the lack or not much noise of sample data needs to be described by subjective uncertainty.On the other hand,fractional differential equation is considered as an effective method to describe the memory and hereditary characteristics of the system.Therefore,for the dynamic system disturbed by this kind of noise,we model it as an uncertain(fractional order)differential equation.In this paper,based on the existing uncertain theory,the first hitting time problems of uncertain systems are discussed,and they are mainly applied in the fields of finance and physics.The main research contents of this thesis are:Different from expect discount criteria,taking the optimistic value of the first hitting time as the objective function,an uncertain optimal optimistic value-based model is proposed.By the definition of ? path,it is transformed into a deterministic optimal control problem.Besides,it proves that the uncertain optimal optimistic value-based model is equivalent to an unconstrained ordinary differential model.The optimal solution of the model is obtained to optimize the optimistic value of the first hitting time.Based on the first hitting time,choose the reaching index as the objective function.An uncertain optimal reaching index model is proposed.By the definition of ? path,it is transformed into a deterministic optimal control problem,it proves that the uncertain optimal control is equivalent to an unconstrained ordinary differential model.The optimal solution of the model is obtained to optimize the reaching index.The extreme value of the solution of Caputo type uncertain fractional differential equation is studied,and two different kinds of uncertain inverse distribution functions of the extreme value are proved.On this basis of extreme value theorems,a numerical algorithm for calculating the uncertain inverse distribution of extreme value is proposed,and a numerical example with analytical expression is provided.The analytical and numerical results of extreme value are compared to demonstrate the effectiveness and accuracy of the numerical algorithm.The first hitting time of the solution of Caputo type uncertain fractional differential equation is studied.Based on the extreme value theorem of solutions of uncertain fractional differential equations,two different kinds of uncertain distribution functions of the first hitting time are proved.Furthermore,a numerical algorithm for solving the uncertain distribution function of the first hitting time is designed by using the predictorcorrector algorithm,and the corresponding numerical examples are given to demonstrate the effectiveness of the numerical algorithm.The first hitting time optimal control problem is applied to practical problems such as portfolio selection in financial field,first order circuit in physics and so on.The sensitivity analysis and guidelines for choosing the parameters are provided for the portfolio model.The uncertain time response is given for the first order circuit model.The extreme value problem of the solution of the uncertain fractional differential equation is applied to the uncertain financial market,and the American options of the uncertain stock model are priced.Meanwhile,the first hitting time of the solution of the uncertain fractional differential equation is applied to the uncertain financial market,and the risk index of the uncertain fractional order mean-reverting model is derived.