An Explicit Single-step Method for Numerical Solution of Optimal Control Problems
Subject Areas : International Journal of Industrial Mathematics
موسی
عبادی
1
(گروه ریاضی، دانشگاه فرهنگیان، تهران، ایران.)
اسفند
ملیح ملکی
2
(گروه ریاضی، دانشگاه پیام نور، تهران، ایران.)
احمدرضا
حقیقی
3
(گروه ریاضی، دانشکده شهید شمسی پور، دانشگاه فنی و حرفهای تهران-ایران.)
علی
عبادیان
4
(گروه ریاضی، دانشگاه ارومیه، ارومیه، ایران.)
Keywords: FBSM, Hybrid methods, stability analysis, OCP,
Abstract :
In this research we used forward-backward sweep method(FBSM) in order to solve optimal control problems. In this paper, one hybrid method based on ERK method of order 4 and 5 are proposed for the numerical approximation of the OCP. The convergence of the new method has been proved .This method indicate more accurate numerical results compared with those of ERK method of order 4 and 5 for solving OCP.
[1] M. Ebadi, New class of hybrid BDF methods for the computation of numerical solution of IVPs, Numer Algor 79 (2018) 179-193.
[2] S. P. Sethi, G. L. Thompson, Optimal Control Theory, Applications to management science and Economics, Kluwer, Boston, 2nd edition, (2000).
[3] S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall, london, (2007).
[4] K. R. Fister, J. C. Panetta, Optimal control Applied to competing chemotherapeutic Cell-Kill strategies, Chapman and Hall, london, (2007).
[5] J. M. Hyman, J. Li, Differential susceptibility and Infectivity Epidemic Models, Mathematical Bioscience and Engineering 11 (2006) 89-100.
[6] G. R. Rose, Numerical Methods for solving Optimal Control Problems, University of Tennesse, Konxville, A Thsis for the master of science Degree (2015).
[7] M. Frego, Numerical Methods for Optimal Control Problems with Application to Autonomous Vehicles, PHD thesis, University of Trento, (2014).
[8] T. R. Goodman, G.N.Lance, Numerical integration of two-point boundary value problems, Math Tables Aids Comput. 12 (1956) 82-86.
[9] D. D. Morrison, J. D. Riley, J. F. Zancanaro, Multiple Shooting Method for two-point boundary value problems, Comm. Acm. 5 (1962) 613-614.
[10] A. V. Rao, A Survey of Numerical Methods for Optimal Control, Advances in the Astronautical Sciences (2010).
[11] l. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, Newyork (1992).
[12] F. Biral, E. Bertolazzi, P. Bosetti, Notes on Numerical Methods for solving Optimal Control Problems, Journal of Industry Applications 5 (2015) 154-166.
[13] I. Grigoryev, S. Mustafina, O. Larin, Numerical Solution of Optimal Control Problems by the Method of Successive Approximations, International Journal of Pure and Applied Mathematics 4 (2016) 617-622.
[14] Z. Wang, Y. Li, An Indirect Method for Inequality Constrained Optimal Control Problems, Science Direct, IFAC Papers online 13 (2017) 4070-4075.
[15] R. Bellman, Dynamic Programming, Princeton University Press, Book (1957).
[16] M. Mcasey, L. Moua, W. Han, Convergence of the Forward-Backward Sweep Method in Optimal Control, Comput Appl. 53 (2012) 207-226.
[17] D. P. Moualeu, M. Weiser, R. Ehrig, P. Deuflhard, Optimal Control for Tuberculosis Model with undetected cases in Cameron, Commun Nonlinear Sci Numer Simulate, Elsiver 20 (2015) 986-1003.
[18] M. Lhous, M. Rachik, H. »aarbi, A. Abdelhak, Discrete Mathematical Modeling and Optimal Control of the marital status, The Monogamous Marriage case, Advances in Difference Equation 39 (2017) 1234-1243.
[19] Q. Chai, W.Wang, A Computational Method for free terminal time Optimal Control Problem governed by Nonlinear time delayed systems, Applied mathematical Modelling 53 (2018) 242-250.
[20] M. K. Jain, Numerical solution of differential equations, 2nd Edition, New Age International publishers 14 (2002) 141-142.
[21] B. Buonomo, On the optimal vaccination strategies for horizontally and vertcally transmitted infectious diseases, Journal of Biological systems 19 (2011) 263-279.