Symplectic multi-level method for solving nonlinear optimal control problem  

作　　者：彭海军 高强 吴志刚 钟万勰 

机构地区：[1]Department of Engineering Mechanics,State Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology [2]School of Aeronautics and Astronautics,State Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology

出　　处：《Applied Mathematics and Mechanics(English Edition)》2010年第10期1251-1260,共10页应用数学和力学（英文版）

基　　金：supported by the National Natural Science Foundation of China(Nos.10632030,10902020,and 10721062);the Research Fund for the Doctoral Program of Higher Education of China(No.20070141067);the Doctoral Fund of Liaoning Province(No.20081091);the Key Laboratory Fund of Liaoning Province of China(No.2009S018);the Young Researcher Funds of Dalian University of Technology(No.SFDUT07002)

摘　　要：By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is proposed.The state and costate variables are approximated by the Lagrange polynomial.The state variables at two ends of the time interval are taken as independent variables.Based on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear equations.Furthermore,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is proposed.When the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency.Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is proposed.The state and costate variables are approximated by the Lagrange polynomial.The state variables at two ends of the time interval are taken as independent variables.Based on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear equations.Furthermore,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is proposed.When the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency.Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.

关 键 词：nonlinear optimal control dual variable variational principle multi-level iteration symplectic algorithm 

分 类 号：O232[理学—运筹学与控制论] TP271[理学—数学]