Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T22:16:01.197Z Has data issue: false hasContentIssue false
  • English
  • Français

SOLVING INFINITE-HORIZON OPTIMAL CONTROL PROBLEMS USING THE HAAR WAVELET COLLOCATION METHOD

Part of: Numerical methods in Fourier analysis Existence theories

Published online by Cambridge University Press:  15 December 2014

ALIREZA NAZEMI  and
NEDA MAHMOUDY
ALIREZA NAZEMI*
Affiliation:
Department of Mathematics, School of Mathematical Sciences, University of Shahrood, PO Box 3619995161-316, Shahrood, Iran email nazemi20042003@yahoo.com
NEDA MAHMOUDY
Affiliation:
Department of Mathematics, Alzahra University, Tehran, Iran email nmahmoudy@yahoo.com
*
✉nazemi20042003@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider infinite-horizon optimal control problems. The main idea is to convert the problem into an equivalent finite-horizon nonlinear optimal control problem. The resulting problem is then solved by means of a direct method using Haar wavelets. A local property of Haar wavelets is applied to simplify the calculation process. The accuracy of the present method is demonstrated by two illustrative examples.

Keywords

infinite horizonoptimal control problemHaar waveletnonlinear programming

MSC classification

Primary: 49J15: Optimal control problems involving ordinary differential equations
Secondary: 65T60: Wavelets 90C30: Nonlinear programming
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 2 , October 2014 , pp. 179 - 191
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Anderson, B. D. O. and Moore, J. B., Linear optimal control (Prentice Hall, Englewood Cliffs, NJ, 1971).Google Scholar
Aubry, S. and Le Daeron, P. Y., “The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states”, Phys. D 8 (1983) 381–422; doi:10.1016/0167-2789(83)90233-6.CrossRefGoogle Scholar
Banks, H. T. and Burns, J. A., “Hereditary control problem: numerical methods based on averaging approximations”, SIAM J. Control Optim. 16 (1978) 169–208; doi:10.1137/0316013.Google Scholar
Blot, J., “Infinite-horizon Pontryagin principles without invertibility”, J. Nonlinear Convex Anal. 10 (2009) 157–176; http://www.ybook.co.jp/online/jncae/vol10/p177.html.Google Scholar
Blot, J. and Michel, P., “First-order necessary conditions for infinite-horizon variational problems”, J. Optim. Theory Appl. 88 (1996) 339–364; doi:10.1007/BF02192175.CrossRefGoogle Scholar
Boggess, A. and Narcowich, F. J., A first course in wavelets with Fourier analysis (Prentice Hall, Englewood Cliffs, NJ, 2009).Google Scholar
Carlson, D. A., Haurie, A. B. and Leizarowitz, A., Infinite-horizon optimal control (Springer, Berlin, 1991).Google Scholar
Chai, Q., Loxton, R., Teo, K. L. and Yang, C., “A max-min control problem arising in gradient-elution chromatography”, Ind. Eng. Chem. Res. 51 (2012) 6137–6144; doi:10.1021/ie202475p.Google Scholar
Dai, R. and Cochran, J. E., “Wavelet collocation method for optimal control problems”, J. Optim. Theory Appl. 143 (2009) 265–278; doi:10.1007/s10957-009-9565-9.Google Scholar
Effati, S., Kamyad, A. V. and Kamyabi-Gol, R. A., “On infinite-horizon optimal control problems”, Z. Anal. Anwend. 19 (2000) 269–278; doi:10.4171/ZAA/950.Google Scholar
Effati, S. and Nazemi, A. R., “A new approach for asymptotic stability of the nonlinear ordinary differential equations”, J. Appl. Math. Comput. 25 (2007) 231–244; doi:10.1007/BF02832349.CrossRefGoogle Scholar
Garg, D., Hager, W. W. and Rao, A. V., “Pseudospectral methods for solving infinite-horizon optimal control problems”, Automatica 47 (2011) 829–837; doi:10.1016/j.automatica.2011.01.085.Google Scholar
Göllmann, L., Kern, D. and Maurer, H., “Optimal control problems with delays in state and control variables subject to mixed control–state constraints”, Optimal Control Appl. Methods 30 (2009) 341–365; doi:10.1002/oca.843.Google Scholar
Karimi, H. R., “A computational method of time-varying state-delayed systems by Haar wavelets”, Int. J. Comput. Math. 83 (2006) 235–246; doi:10.1007/s10883-005-4172-z.Google Scholar
Lin, Q., Loxton, R. and Teo, K. L., “The control parameterization method for nonlinear optimal control: A survey”, J. Ind. Manag. Optim. 10 (2014) 275–309; doi:10.3934/jimo.2014.10.275.Google Scholar
Lin, Q., Loxton, R., Teo, K. L. and Wu, Y. H., “A new computational method for a class of free terminal time optimal control problems”, Pac. J. Optim. 7 (2011) 63–81; http://www.ybook.co.jp/online2/oppjo/vol7/p63.html.Google Scholar
Loxton, R., Lin, Q. and Teo, K. L., “Minimizing control variation in nonlinear optimal control”, Automatica 49 (2013) 2652–2664; doi:10.1016/j.automatica.2013.05.027.Google Scholar
Marcus, M. and Zaslavski, A. J., “The structure of extremals of a class of second order variational problems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 593–629; doi:10.1016/S0294-1449(99)80029-8.Google Scholar
Marzban, H. R. and Razaghi, M., “Rationalized Haar approach for nonlinear constrained optimal control problems”, Appl. Math. Model. 34 (2010) 174–183; doi:10.1016/j.apm.2009.03.036.Google Scholar
Ohkita, M. and Kobayashi, Y., “An application of rationalized Haar functions to solution of linear differential equations”, IEEE Trans. Circuits Syst. 33 (1986) 853–862; doi:10.1016/0378-4754(88)90055-9.CrossRefGoogle Scholar
Plant, D., Smith, D. and Darrel, P., The Lingo programmer’s reference (Ventana Communications Group, Research Triangle Park, NC, 1997).Google Scholar
Razzaghi, M. and Ordokhani, Y., “Solution of differential equations via rationalized Haar functions”, Math. Comput. Simulation 56 (2001) 235–246; doi:10.1016/S0378-4754(01)00278-6.Google Scholar
Rudin, W., Real and complex analysis (McGraw-Hill, New York, 1966).Google Scholar
Smirnov, G. V., “Transversality condition for infinite-horizon problems”, J. Optim. Theory Appl. 88 (1996) 671–688; doi:10.1007/BF02192204.CrossRefGoogle Scholar
Ye, J. J., “Non-smooth maximum principle for infinite-horizon problems”, J. Optim. Theory Appl. 76 (1993) 485–500; doi:10.1007/BF00939379.Google Scholar
Zaslavski, A. J., “Turnpike results for a discrete-time optimal control system arising in economic dynamics”, Nonlinear Anal. 67 (2007) 2024–2049; doi:10.1016/j.na.2006.08.029.Google Scholar