In this paper, Homotopy perturbation method is used to find the approximate solution of the optimal control of linear systems. In this method the initial approximations are freely chosen, and a Homotopy is constructed with an embedding parameter , which is considered as a “small parameter”. Some examples are given in order to find the approximate solution and verify the efficiency of the proposed method.
KeywordsHomotopy Perturbation MethodOptimal Control ProblemHamilton System
M. Itik, M. U. Salamci and S. P. Banksa, “Optimal Control of Drug Therapy in Cancer Treatment,” Nonlinear Analysis, Vol. 71, No. 12, 2009, pp. e1473-e1486. doi:10.1016/j.na.2009.01.214
W. L. Garrard and J. M. Jordan, “Design of Nonlinear Automatic Flight Control Systems,” Automatic, Vol. 13, No. 5, 1977, pp. 497-505. doi:10.1016/0005-1098(77)90070-X
S. Wei, M. Zefran and R. A. DeCarlo, “Optimal Control of Robotic System with Logical Constraints: Application to UAV Path Planning,” Proceedings of the IEEE International Conference on Robotic and Automation, Pasadena, 19-23 May 2008, pp. 176-181.
I. Chryssoverghi, J. Coletsos and B. Kokkinis, “Approximate Relaxed Descent Method for Optimal Control Problems,” Control and Cybernetics, Vol. 30, No. 4, 2001, pp. 385-404.
D. E. Kirk, “Optimal Control Theory: An Introduction,” Prentice-Hall, Upper Saddle River, 1970.
J. C. Dunn, “On L2 Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems,” SIAM Journal of Continues Optimal, Vol. 34, No. 4, 1996, pp. 1270-1290. doi:10.1137/S0363012994266127
R. W. Beard, G. N. Saridis and J. T. Wen, “Approximate Solutions to the Time-Invariant Hamilton-Jacobi-Bellman Equation,” Optimal Theory Application, Vol. 96, No. 3, 1998, pp. 589-626. doi:10.1023/A:1022664528457
I. Chryssoverghi, I. Coletsos and B. Kokkinis, “Discretization Methods for Optimal Control Problems with State Constraints,” Journal of Computational and Applied Mathematics, Vol. 19, No. 1, 2006, pp. 1-31. doi:10.1016/j.cam.2005.04.020
A. V. Kamyad, M. Keyanpour and M. H. Farahi, “A New Approach for Solving of Optimal Nonlinear Control Problems,” Applied Mathematics. Computers, Vol. 187, No. 2, 2007, pp. 1461-1471. doi:10.1016/j.amc.2006.09.051
S. Effati, M. Janfada and M. Esmaeili, “Solving the Optimal Control Problem of the Parabolic PDEs in Exploitation of Oil,” Journal of Mathematical Analysis and Applications, Vol. 340, No. 1, 2008, pp. 606-620. doi:10.1016/j.jmaa.2007.08.037
O. S. Fard and H. A. Borzabadi, “Optimal Control Problem, Quasi-Assignment Problem and Genetic Algorithm,” Proceedings of World Academy of Science, Engineering and Technology, Vol. 21, 2007, pp. 70-43.
K. L. Teo, C. J. Goh and K. H. Wong, “A Uni?ed Computational Approach to Optimal Control Problem,” Longman Scienti?c and Technical, Harlow, 1991.
H. Hashemi Mehne and A. Hashemi Borzabadi, “A Numerical Method for Solving Optimal Control Problem Using State Parametrization,” Numerical Algorithms, Vol. 42, No. 2, 2006, pp. 165-169. doi:10.1007/s11075-006-9035-5
G. N. Elnagar, “State-Control Spectral Chebyshev Parameterization for Linearly Constrained Quadratic Optimal Control Problems,” Computational Applied Mathematics, Vol. 79, No. 1, 1997, pp. 19-40. doi:10.1016/S0377-0427(96)00134-3
J. Vlassenbroeck and R. V. Dooren, “A Chebyshev Technique for Solving Nonlinear Optimal Control Problems,” IEEE Transactions on Automatic Control, Vol. 33, No. 4, 1998, pp. 333-340. doi:10.1109/9.192187
H. R. Sirsena and K. S. Tan, “Computation of Constrained Optimal Controls Using Parameterization Techniques,” IEEE Transactions on Automatic Control, Vol. 19, No. 4, 1974, pp. 431-433.
H. P. Hua, “Numerical Solution of Optimal Control Problems,” Optimal Control Applications and Methods, Vol. 21, No. 5, 2000, pp. 233-241. doi:10.1002/1099-1514(200009/10)21:5 3.0.CO;2-B
V. V. Dikusar, M. Kosh’ka and A. Figura, “Parametric Continuation Method for Boundary-Value Problems in Optimal Control,” Differentsial/cprime nye Uravneniya, Vol. 37, No. 4, 2001, pp. 453-457.
M. Weiser, “Function Space Complementarity Methods for Optimal Control Problems,” Dissertation Eingereicht am Fachbereich Mathematik und Informatik der Freien Universitat, Berlin, 2001.
M. E. Lahaye, “Solution of System of Transcendental Equations,” Académie Royale de Belgique. Bulletin de la Classe des Sciences, Vol. 5, 1948, pp. 805-822.
D. F. Davidenko, “Solution of System of Transcendental Equations,” Dokl. Akad. Nauk, Vol. 88, No. 4, 1953, pp. 601-602.
D. F. Davidenko, “Approximate Solution of Systems of Nonlinear Equations,” Ukr. Mat. Zh., Vol. 5, No. 2, 1953, pp. 196-206.
V. E. Shamanskii, “Metody Chislennogo Resheniya Kraevykh Zadach na EtsVM (Numerical Methods for the Solution of Boundary Value Problems on Computer),” Naukova Dumka, Kiev, 1966.
S. Roberts and J. S. Shipman, “Continuation in Shooting Methods for Two-Point Boundary Value Problems,” Journal of Mathematical Analysis and Applications, Vol. 18, No. 1, 1967, pp. 45-58. doi:10.1016/0022-247X(67)90181-3
E. I. Grigolyuk and V. I. Shalashilin, “Problemy Nelineinogo Deformirovaniya (Problems of Nonlinear Deformation), Nauka, Moscow, 1988.
V. I. Shalashilin and E. B. Kuznetsov, “Metod Prodolzhneiya Resheniya po Parametru i Nailuchshaya Parametrizatsiya (The Homotopy Method of Continuation and Best Parametrization),” Editorial URSS, Moscow, 1999.
S. N. Avvakumov, “Smooth Approximation of Convex Compacta,” Trudy Instituta. Matematiki. i Mekhaniki. UrO RAN, Ekaterinburg, Vol. 4, 1996, pp. 184-200.
E. L. Allgower and K. Georg, “Introduction to Numerical Continuation Methods,” SIAM, Berlin, 1990.
S. A. Yousefi, M. Dehghan and A. Lotfi, “Finding Optimal Control of Linear Systems via He’s Variational Iteration Method,” Computational Mathematics, Vol. 87, No. 5, 2010, pp. 1042-1050.
G. L. Liu, “New Research Directions in Singular Perturbation Theory: Artificial Parameter Approach and Inverse Perturbation Technique,” Proceeding of the 7th Conference of the Modern Mathematics and Mechanics, Shanghai, September 1997, pp. 47-53.
S. Abbasbandy, “Homotopy Perturbation Method for Quadratic Riccati Differential Equation and Comparision with Adomian’s Decomposition Method,” Applied Applied Mathematics and Computation, Vol. 172, 2006, pp. 482-490.
D. Ganji, H. Tari and M. Bakhshi, “Variational Iteration Method and Homotopy Perturbation Method for Nonlinear Evalution Equations,” Computers & Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 1018-1024. doi:10.1016/j.camwa.2006.12.070
J.-H. He, “A Coupling Method for a Homotopy Technique and a Perturbation Technique for Nonlinear Problems,” International Journal of Non-Linear Mechanics, Vol. 35, No. 1, 2000, pp. 37-43.
J.-H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5
J.-H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 87-88. doi:10.1016/j.physleta.2005.10.005
J.-H. He, “Homotopy Perturbation Technique,” Applied Mathematics and Computation, Vol. 178, No. 2, 1997, pp. 257-262.