A New Analytical Method for Solving Hamilton-jacobi-bellman Equation

Volume 11, Issue 4, pp 252 - 263 http://dx.doi.org/10.22436/jmcs.011.04.01

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2942 Downloads
• 4010 Views

Authors

M. Matinfar - Department of Mathematics, Faculty of Science, University of Mazandaran, Babolsar, Iran. M. Saeidy - Department of Mathematics, Faculty of Science, University of Mazandaran, Babolsar, Iran.

Abstract

In this paper, we apply a modification of variational iteration method using He's polynomials for a class of nonlinear optimal control problems which are converted to the Hamilton-Jacobi-Bellman equations (HJB) and present a convergence theorem of the method. The proposed modification is made by introducing He's polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for using in these problems. Some examples are given to demonstrate the simplicity and efficiency of the proposed method.

Share and Cite

Keywords

• Optimal control problem
• Homotopy perturbation method
• Variational iteration method
• Numerical solution.

MSC

•  93B40
•  49M15
•  49L20

References

• [1] M. Itik, M. U. Salamci, S. P. Banksa, Optimal control of drug therapy in cancer treatment, Nonlinear Analysis, 71(12) (2009), 1473-1486.
  • View Article
  • Google Scholar


• [2] W. L. Garrard, J. M. Jordan , Design of nonlinear automatic flight control systems , Automatica, 13(5) (1977), 497-505.
  • View Article
  • Google Scholar


• [3] S. Wei, M. Zefran, R. A. DeCarlo, Optimal control of robotic system with logical constraints: application to UAV path planning, In Proceeding(s) of the IEEE International Conference on Robotic and Automation, Pasadena, CA, USA (2008)
  • View Article
  • Google Scholar


• [4] O. Stryk, R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37(1-4) (1992), 357-373.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] C. J. Goh, K. L. Teo, Control parameterization: a unified approach to optimal control problem with general constraints, Automatica, 24(1) (1988), 3-18.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] R. Bellman, On the theory of dynamic programming, Proceedings of the National Academy of Sciences, USA, 38(8) (1952), 716-719.
  • View Article
  • Google Scholar


• [7] L. S. Pontryagin, Optimal control processes, Uspekhi Matematicheskikh Nauk, 14 (1959), 3-20.


• [8] R. W. Beard, G. N. Saridis, J. T. Wen, Galerkin approximations of the generalized Hamilton- Jacobi-Bellman equation, Automatica, 33(12) (1997), 2159-2177.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] G. Y. Tang, Suboptimal control for nonlinear systems: a successive approximation approach, System and Control Letters, 54(5) (2005), 429-434.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] G. Y. Tang, H. P. Qu, Y. M. Gao, Sensitivity approach of suboptimal control for a class of nonlinear systems, Journal of Ocean University of Qingdao, 32(4) (2002), 615-620.
  • Google Scholar


• [11] M. A. Abdou, A. A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equations, J. Comput. Appl. Math., 181 (2005), 245-251.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. R. Caputo, Foundations of dynamic economic analysis: optimal control theory and applications, Cambridge University press, (2005)
  • View Article
  • Google Scholar
  • MATH


• [13] A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Solitons Fractals, 39 (2009), 1486-1492.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] A. Ghorbani, J. Saberi-Nadjafi, He's homotopy perturbation method for calculating adomian polynomials, Int. J. oF Non. Sci. and Num. Simul., 8 (2007), 229-232.
  • View Article
  • Google Scholar


• [15] J. H. He, Homotopy perturbation technique, Comput. Math. in Appl. Mech. and Eng., 178(3-4) (1999), 257-62.
  • View Article
  • Google Scholar


• [16] J. H. He, Variational iteration method:Some recent results and new interpretations, J. Comput. Appl. Math., 207 (2007), 3-7.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] M. Inokuti, et al., General use of the Lagrange multiplier in non-linear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, (1978), 156-162.
  • Google Scholar


• [18] S. T. Mohyud-Din, M. A. Noor, KI. Noor, Variational Iteration Method for Burgers' and Coupled Burgers' Equations Using He's Polynomials, Zeitschrift Fur Naturforschunge Section A-A J. oF Phy. Sci., 65 (2010), 263-267.
  • View Article
  • Google Scholar


• [19] M. Matinfar, M. Saeidy, The Homotopy perturbation method for solving higher dimensional initial boundary value problems of variable coefficients, W. J. of Model. and Simu., 5 (2009), 75-83.
  • Google Scholar


• [20] A. M. Wazwaz , The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. and Math. with Appl., 54 (2007), 926-932.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] A. M. Wazwaz, The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations, J. Comput. Appl. Math., 207 (2007), 18-23.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. A. Yousefi, M. Dehghan, A. Lotfi, Finding the optimal control of linear systems via He's variational iteration method, International Journal of Computer Mathematics, (2010), 1042-1050.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH