Research Based On Rational Homotopy Perturbation Method For Solving Optimal Control Problems

In recent years,many experts and scholars have proposed various methods and strategies for solving optimal control problems.Among them,the more famous are the variational iteration method,the dynamic programming method,the perturbation method with small parameters,and the homotopy analysis method which has artificial parameters,and the most widely used homotopy perturbation method.This paper mainly studies using rational homotopy perturbation method to solve linear and nonlinear optimal control problems.The rational homotopy perturbation method is an improvement and enhancement of the homotopy perturbation method.The method uses the form of two power series quotients to promote the homotopy perturbation method,introducing adjustment parameters ?_i and ?_i.The calculation accuracy is greatly improved,and it can be found that very good results can be achieved through a few iterations.(1)For the complexity of the rational homotopy perturbation method fitting and solving parameters.This article also provides a new idea,that is,the rational biparameter homotopy perturbation method,which introduces two parameters to solve the differential equation,which is verified by examples,which is superior to homotopy perturbation method and does not require fitting parameters.Other,in order to improve accuracy,there is a new combination of Laplace-Pade transform and the rational biparameter homotopy perturbation method.Through comparison found that the accuracy of the Laplace-Pade rational biparameter homotopy perturbation method is much higher than the rational biparameter homotopy perturbation method.(2)By using the Pontryagin minimum principle,we use the rational homotopy perturbation method to solve two examples of first-order linear systems and an example of a two-order linear system,and compared with the homotopy perturbation method and the homotopy analysis method.It is found that the rational homotopy perturbation method is a method with the least number of iterations and is very accurate,which almost close to the exact solution.Therefore,the rational homotopy perturbation method is very effective in solving the optimal control problem of linear systems.(3)Using rational homotopy perturbation method to solve the nonlinear optimal control problems.Need to use He'polynomials method to deal with the nonlinear term firstly,determine the initial value,then find the sub-control to minimize the performance index.Finally,an example is given to illustrate the specific application of the rational homotopy perturbation method in its nonlinear optimal control system.And compared with the homotopy perturbation method,it can be found that the rational homotopy perturbation method is superior.