Symplectic multi-level method for solving nonlinear optimal control problem

• Published in: Applied mathematics and mechanics Vol. 31; no. 10; pp. 1251 - 1260
• Main Author: 彭海军 高强 吴志刚 钟万勰
• Format: Journal Article
• Language: English
• Published: Heidelberg Shanghai University Press 01.10.2010; Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, Liaoning Province, P. R. China%School of Aeronautics and Astronautics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, Liaoning Province, P. R. China
• Subjects: Algorithms; Applications of Mathematics; Approximation; Classical Mechanics; Fluid- and Aerodynamics; Low level; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; Optimal control; Partial Differential Equations; 哈密顿系统; 多层次; 多项式近似; 拉格朗日插值; 最优控制问题; 状态变量; 非线性方程组; 非线性系统; variational principle; O231.2; 34B15; multi-level iteration; dual variable; O241.81; nonlinear optimal control; symplectic algorithm
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-010-1358-6

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.