Research ArticleOpen AccessGoogle Scholar indexed
Parametric Iteration Method for Solving Linear Optimal Control Problems
Department of Mathematics, Payame Noor University, Tehran, Iran
Department of Mathematics, Payame Noor University, Mashhad, Iran
- 1 Department of Mathematics, Payame Noor University, Tehran, Iran
- 2 Department of Mathematics, Payame Noor University, Mashhad, Iran
Applied Mathematics·Volume 03 (2012)·Pages 1059–1064·Published 27 September 2012·DOI10.4236/am.2012.39155
Copy link · social · email
Abstract
This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.
KeywordsParametric Iteration MethodOptimal Control ProblemPontryagin’s Maximum PrincipleHe’s Variational Iteration Method
- L. Ntogramatzidis and A. Ferrante, “On the Solution of the Riccati Differential Equation Arising from the LQ Optimal Control Problem,” Systems & Control Letters, Vol. 59, No. 2, 2010, pp. 114-121. doi:10.1016/j.sysconle.2009.12.006
- P. Williams, “A Gauss-Lobatto Quadrature Method for Solving Optimal Control Problems,” ANZIAM Journal, Vol. 47, 2006, pp. C101-C115.
- M. Yamaguti and S. Ushiki, “Chaos in Numerical Analysis of Ordinary Differential Equations,” Physica D: Nonlinear Phenomena, Vol. 3, No. 3, 1981, pp. 618-626. doi:10.1016/0167-2789(81)90044-0
- C. K. Chui and G. Chen, “Linear Systems and Optimal Control,” Springer-Verlag, Berlin, Heidelberg, 1989. doi:10.1007/978-3-642-61312-8
- S. A. Yousefi, M. Dehghan and A. Lotfi, “Finding the Optimal Control of Linear Systems via He’s Variational Iteration Method,” International Journal of Computer and Mathematics, 2009.
- A. Ghorbani, “Toward a New Analytical Method for Solving Nonlinear Fractional Differential Equations,” Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 49-50, 2008, pp. 4173-4179. doi:10.1016/j.cma.2008.04.015
- J. H. He, “Variational Iteration Method—A Kind of Nonlinear Analytical Technique: Some Examples,” International Journal of Non-Linear Mechanics, Vol. 34, 1999, pp. 699-708.
- A. Ghorbani and J. S. Nadjafi, “A Piecewise-Spectral Parametric Iteration Method for Solving the Nonlinear Chaotic Genesio System,” Mathematical and Computer Modeling, Vol. 54, No. 1-2, 2011, pp. 131-139. doi:10.1016/j.mcm.2011.01.044