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The Scientific World Journal
Volume 2013, Article ID 306237, 9 pages
Research Article

Numerical Solution of Some Types of Fractional Optimal Control Problems

1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany
3Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA

Received 11 September 2013; Accepted 30 September 2013

Academic Editors: C. Li, F. Liu, R. Magin, A. Sikorskii, and S. B. Yuste

Copyright © 2013 Nasser Hassan Sweilam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present two different approaches for the numerical solution of fractional optimal control problems (FOCPs) based on a spectral method using Chebyshev polynomials. The fractional derivative is described in the Caputo sense. The first approach follows the paradigm “optimize first, then discretize” and relies on the approximation of the necessary optimality conditions in terms of the associated Hamiltonian. In the second approach, the state equation is discretized first using the Clenshaw and Curtis scheme for the numerical integration of nonsingular functions followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Two illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.