Document Type : Research Article
Authors
1 Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
Abstract
In this paper, a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly. First, the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs). Then, the unknown functions are approximated by the hybrid functions, including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method. Also, two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method. The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method, which determines the solution of the problem. Error estimates for the hybrid function approximation, fractional operators and, the proposed method are provided. Finally, the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems.
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Main Subjects
begin{thebibliography}{99}
Kilbas A. A., Srivastava H. M., Trujillo J. J., (2006). “Theory and Applications of Fractional Differential Equations”, 204 (North-Holland Mathematics Studies), Elsevier Science Inc., New York, NY.
Samko S. G., Kilbas A. A., Marichev O. I., (1993). “Fractional Integrals and Derivatives: Theory and Applications”, Switzerland: Gordon and Breach Science Publishers.
Agrawal O. P., (2004). “A general formulation and solution scheme for fractional optimal control problems”, Nonlinear Dynamics, 38 (1-4), 323-337.
Agrawal O. P., (2008). “A quadratic numerical scheme for fractional optimal control problem”, Journal of Dynamic Systems, Measurement, and Control, 130(1), 011010-011010-6.
Kirk D. E., (1970). “Optimal Control Theory: An Introduction”, Dover Publication, New York.
Baleanu D., Defterli O., Agrawal O.P., (2009). “A central difference numerical scheme for fractional optimal control problems”, Journal of Vibration and Control, 15(4), 583-597.
Bhrawy A. H., Ezz-Eliden S. S., Doha E.H., Abdelkawy M.A., Baleanu D., (2017). “Solving fractional optimal control problems with in Chebyshev-Legendre operational technique”, International Journal of Control, 90 (6), 1230-1244.
Ticaud C., Chen Y. Q., (2010). “An approximation method for numerically solving fractional optimal control problems of general form”, Computers and Mathematics with Applications, 59(5), 1644-1655.
Shen J., Cao J., (2012). “Necessary and sufficient conditions for consensus of delayed fractional-order systems”, Asian Journal of Control, 14(6), 1690-1697.
Gelfand M., Fomin S. V., (1963). “Calculus of Variations”, Prentice-Hall.
Mamehrashi K., Nemati A., (2019). “A new approach for solving infinite horizon optimal control problems using Laguerre functions and Ritz spectral method”, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2019.1628949.
Doha E., Bhrawy A., Ezz-Eldien S., (2011). “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations”, Applied Mathematical Modelling, 35(12), 5662–5672.
Esmaeili S., Shamsi M., (2011). “A pseudo-spectral scheme for the approximate solution of a family of fractional differential equation”, Communications in Nonlinear Science and Numerical Simulation, 16 (9), 3646– 3654.
Sweilam N.H., Al-Ajami T.M., (2015). “Legendre spectral-collocation method for solving some types of fractional optimal control problems”, Journal of Advanced Research, 6 (3), 393-403.
Nemati A., (2017). “Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix”, International Journal of Control, https://doi.org/10.1080/00207173429.2017.1367.
Ejlali N., Hosseini S.M., Yousefi S.A., (2018). “B-spline spectral method for constrained fractional optimal control problems”, Mathematical Methods in the Applied Sciences, 41(14), 5466-5480.
Hoseini S. M., Marzban H. R., Razzaghi M., (2009). “Solution of Volterra population model via Block-pulse functions and Lagrange-interpolating polynomials”, Mathematical Methods in the Applied Sciences, 32, 127–134.
Mashayekhi S., Razzaghi M., (2015). “Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation”, Mathematical Methods in the Applied Sciences, 39, 353-365.
Razzaghi M., Marzban H. R., (2000). “Direct method for variational problems via hybrid of Block-pulse and Chebyshev functions”, Mathematical Problems in Engineering, 6, 85–97.
bibitem{21}
Singh V. K., Pandey R. K., Singh S., (2010). “A stable algorithm for Hankel transforms using hybrid of Block-pulse and Legendre polynomials”, Computer Physics Communications, 181(1) 1–10.
bibitem{22}
Dadkhah M., Farahi M. H., Heydari A., (2018). “Numerical solution of time delay optimal control problems by hybrid of block-pulse functions and Bernstein polynomials”, IMA Journal of Mathematical Control and Information, 35 (2), 451–477.
bibitem{23}
Dehestani H., Ordokhani Y., Razzaghi M., (2019). “Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays”, International Journal of Numerical Modelling, e2606. https://doi.org/10.1002/jnm.2606.
Mashayekhi S., Ordokhani Y., Razzaghi M., (2012). “Hybrid functions approach for nonlinear constrained optimal control problems”, Communications in Nonlinear Science and Numerical Simulation, 17, 1831-1843.
Haddadi N., Ordokhani Y., Razzaghi M., (2012). “Optimal control of delay systems by using a hybrid functions approximation”, Journal of Optimization Theory and Applications, 153, 338–356.
Podlubny I., (1999). “Fractional Differential Equations: an Introduction to Fractional Derivatives Fractional Differential Equations, to Methods of their Solution and some of their Applications”, Academic Press, San Diego.
Zhou Y., (2014). “Basic Theory of Fractional Differential Equations”, Wo. I. rld. Scientific Publishing Co. Pte.Ltd.
bibitem{28}
Viola C., (2016). “An Introduction to Special Functions”. Springer International Publishing, Switzerland.
Kreyszig E., (1989). “Introductory Functional Analysis with Applications”, vol. 81, John Wiley & Sons, New York, NY, USA.
Mohammadi F., Moradi L., Baleanu D., Jajarmi A., (2017). “A hybrid functions numerical scheme for fractional optimal control problem: Application to nonanalytic dynamic systems”, Journal of Vibration and Control, 1-14.
Canuto C., Quarteroni A., Hussaini M. Y., Zang T. A., (2006). “Spectral Methods: Fundamentals in Single Domains”, Springer-Verlag Berlin Heidelberg.
Nemati A., Yousefi S. A., (2016). “A numerical method for solving fractional optimal control problems using Ritz method”, Journal of Computational and Nonlinear Dynamics, 11, 1-7.
Lotfi A., Yousefi S. A., (2014). “Epsilon-Ritz method for solving a class of fractional constrained optimization problems”, Journal of Optimization Theory and Applications, 163(3) 884-899.
Ezz-Eldien S. S., Doha E.H., Baleanu D., Bhrawy A.H., (2015).“A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems”, Journal of Vibration and Control, 23 (1) 16 – 30.
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