A Class Of Linear Quadratic Optimal Control Problems Under Stochastic Disturbances

In recent decades,optimal control theory has been widely used in industrial production,drug therapy,finance and other fields.Linear quadratic optimal control model mainly studies the control problem of the system state in the form of linear quadratic form,which is an important class of optimal control problems.Its greatest advantage is that it can calculate accurate optimal control.However,in the process of system operation,the system will inevitably be disturbed by some stochastic factors.Therefore,the deterministic linear quadratic optimal control problem can no longer accurately describe the operation process of the system.In order to better deal with the interference of random factors,stochastic linear quadratic optimal control problem was discovered.In the case of stochastic dynamic systems,the optimal control of the linear quadratic optimal control problem can also be calculated,but the optimal control obtained depends on a Riccati differential equation,which results in the analytical control equation being very complex.Therefore,the optimal control obtained from the stochastic linear quadratic optimal control problem cannot be applied to practical problems.This paper presents a stochastic linear quadratic parameter optimal control problem and a control parameterization method for solving the parameters.By introducing the control parameterization method,the form of optimal control is greatly simplified and some errors are caused.Therefore,on the basis of the control parameterization method,a control parameter segmentation method is proposed to reduce the errors.The research content of this paper is mainly divided into the following three aspects:(1)The stochastic linear quadratic optimal control model is constructed.According to the optimality equation of the stochastic control problem,the analytical forms of the optimal control and optimal value of the stochastic linear quadratic optimal control model are derived.Secondly,the Riccati differential equations that depend on the analytical forms of the optimal control and optimal value are studied,and the properties of the solutions of the Riccati differential equations are analyzed.(2)Based on the research of stochastic linear quadratic optimal control problem,the stochastic linear quadratic parameter optimal control model is constructed and the control parameterization method is proposed to solve the control parameters.Then,the form of parameter optimal control,the form of optimal value under parameter control and the parameter differential equation on which it depends are derived,and the properties of the solution of the parameter differential equation are analyzed.Finally,in order to reduce the error between the optimal value and the optimal value under parameter control,the control parameter segmentation method is proposed to increase the number of control parameters,and then the form of piecewise control parameters and the form of optimal value under piecewise control parameters are analyzed.(3)Establish an inventory control problem interfered by random factors,and solve the optimal control equation and the magnitude of the optimal value of the inventory problem.Then,according to the above-mentioned control parameterization method,the control forms for one control parameter,two control parameters and five control parameters are solved respectively,and the optimal values under the corresponding control forms of different inventories are calculated.Finally,the accurate optimal control and optimal value are compared with parameterized optimal control and optimal value,and the feasibility and effectiveness of the proposed control parameterization method are verified.