توابع کاردینال هرمیت و کاربرد آن‌ها در حل مسأله کنترل بهینه کسری تأخیری

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشجوی دکتری دانشکده علوم ریاضی، دانشگاه الزهرا(س)

2 گروه ریاضی، دانشکده علوم ریاضی، دانشگاه الزهرا، تهزان، ایران

چکیده

در این مقاله، یک روش عددی جدید برای حل مسأله کنترل بهینه کسری با تأخیر در زمان ارائه شده است. انتگرال کسری از نوع
ریمان-لیوویل و مشتق کسری از نوع کاپوتو می‌باشد. در این روش، از توابع کاردینال هرمیت به­عنوان توابع پایه برای تقریب توابع استفاده می­شود. در ادامه، ماتریس عملیاتی انتگرال کسری و تأخیری به­دست می‌آید و آن‌ها برای حل مسأله کنترل بهینه به‌کار برده می­شوند. با استفاده از روش هم­مکانی، مسأله مورد مطالعه به یک دستگاه معادلات جبری منجر شده که با استفاده از روش تکراری نیوتن جواب مساله محاسبه می­شود. در پایان، با ارائه مثال­های عددی کارایی روش بررسی شده است.

کلیدواژه‌ها


عنوان مقاله [English]

Cardinal Hermit functions and their application in solving the time-delay fractional optimal control problems

نویسندگان [English]

  • F. S. Yosefi 1
  • Yadollah Ordokhani 2
1 Department of Mathematics,, Faculty of Mathematical Sciences,, Alzahra University, Tehran, Iran
2 Department of Mathematics,, Faculty of Mathematical Sciences,, Alzahra University, Tehran, Iran
چکیده [English]

In this paper, a new numerical method for solving the fractional optimal control problem of the time delay is  presented. The fractional integral and the fractional derivative are the Riemann-Liouville type and the Caputo type, respectively. In this method, the cardinal Hermite functions are used as a basis to approximate functions. Moreover, we obtain the fractional and delay integral operational matrices and use them to solve this optimal control problem. Using the collocation method, the problem leads to a system of algebraic equations, that is solved by Newton's        iterative method. Finally, numerical examples are presented to investigate the efficiency of this method.
 

کلیدواژه‌ها [English]

  • Fractional optimal control problem of time delay
  • Operational matrices
  • Cardinal Hermite functions
  [1]      D. A. Benson, M. M. Meerschaert, J. Revielle, “Fractional calculus in hydrologicmodeling: a numerical perspective,” Adv. Water Resour, vol. 51, pp. 479–497, 2013.##
  [2]      J. K. Popovic, D.T. Spasic, J. Tosic, J. L Kolarovic, and R. Malti, “Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia,” Commun. Nonlinear Sci. Numer. Simul., vol. 22, pp. 451–471, 2015.##
  [3]      D. Sierociuk, A. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, and T. Skovranek, “ Modelling heat transfer in heterogeneous madia using fractional calculus,” Phil. Trans. R. Soc. A., vol. 371, pp.   2013-2046, 2013.##
  [4]      S. Larsson, M. Racheva, and F. Saedpanah, “Discontinuous Galerkin method for an          integro-differential equation modeling dynamic fractional order viscoelasticity,” Comput. Method. Appl. Mech. Eng., vol. 283, pp. 196–209, 2015.##
  [5]      Y. Jiang and X. Wang, “On a stochastic heat equation with first order fractional noises and applications to finance,” J. Math. Anal. Appl., vol. 396, pp. 656–669, 2012.##
  [6]      G. Bohannan, “Analog fractional order controller in temperature and motor control applications,” J. Vib. Contr., vol. 14, pp. 1487–1498, 2008.##
  [7]      O. P. Agrawal, “A formulation and numerical scheme for fractional optimal control problems,” J. Vib. Control, vol. 14, pp. 1291–1299, 2008.##
  [8]      A. Lotfi, S. A. Yousefi, and M. Dehghan, “Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operationalmatrix and the Gauss quadrature rule,” J. Comput. Appl. Math., vol. 250, pp. 143–160, 2013.##
  [9]      O. P. Agrawal, “Fractional optimal control of a distributed system using eigenfunctions,” ASME. J. Comput. Nonlinear Dyn., vol. 3, pp. 2- 6, 2008.##
[10]      M. Jamshidi and C. M.  Wang, “A computational algorithm for large-scale nonlinear time-delay systems,” IEEE Trans. Syst. Man Cybern., vol. 14, pp. 2–9, 1984.##
[11]      A. H. Bhrawy and S. S. Ezz-Eldien, “A new Legendre perational technique for delay fractional optimal control problems,” Calcolo, vol. 53(4), pp. 521–543, 2016.##
[12]      P. Rahimkhani, Y. Ordokhani, and E. Babolian, “An efficient approximate method for solving delay fractional optimal control problems,” Nonlinear Dynamics, vol. 86(3), pp. 1649–1661, 2016.##
[13]      L. Moradi, F. Mohammadi, and D. Baleanu, “A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets,” J. Vib. Contr., vol. 25, pp. 1–15, 2018.##
[14]      K. Rabiei, Y. Ordokhani, and E. Babolian, “Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems,” J. Vib. Contr., vol. 24, pp.     3370-3383, 2018.##
[15]      E. Safaie, M. H. Farahi, and M. F. Ardehaie, “An approximate method for numerically solving      multi-dimensional delay fractional optimal control problems by Bernstein polynomials,” Comput. Appl.  Math., vol. 34, pp. 831–846, 2015.##
[16]      B. Han and Q. Jiang, “Multiwavelets on the interval,” Appl. Comput. Harmon. Anal., vol. 12, pp. 100–127, 2002.##
[17]      E. Keshavarz, Y. Ordokhani, and M.  Razzaghi, “Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations,” Appl. Math. Model., vol. 38(24), pp. 6038–6051, 2014.##
[18]      E. Ashpazzadeh, M. Lakestani, and M. Razzaghi, “Nonlinear Constrained Optimal Control Problems and Cardinal Hermite Interpolant Multiscaling Functions,” Asian. J. Control., vol. 20, pp. 558–567, 2018.##
[19]      H. R. Marzban and M. Razzaghi, “Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials,” J. Frankl. Inst., vol. 341, pp. 279–293, 2004.##
[20]      M. H. Farahi and M.  Dadkhah, “Solving nonlinear time delay control systems by Fourier series,” Int. J. Eng. Res. Appl., vol. 5, pp. 217–226, 2014.##