Research Papers

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

[+] Author and Article Information
Sohrab Effati

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: s-effati@um.ac.ir

Seyed Ali Rakhshan

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: seyedalirakhshan@yahoo.com

Samane Saqi

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: samane.saqi@mail.um.ac.ir

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 27, 2017; final manuscript received April 4, 2018; published online May 2, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(6), 061007 (May 02, 2018) (10 pages) Paper No: CND-17-1522; doi: 10.1115/1.4039900 History: Received November 27, 2017; Revised April 04, 2018

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

Copyright © 2018 by ASME
Topics: Optimal control
Your Session has timed out. Please sign back in to continue.


Bagley, R. L. , and Torvik, P. J. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27(3), pp. 201–210. [CrossRef]
Baillie, R. T. , 1996, “ Long Memory Processes and Fractional Integration in Econometrics,” J. Econometrics, 73(1), pp. 5–59. [CrossRef]
Carpinteri, A. , and Mainardi, F. , eds., 1997, Fractals and Fractional Calculus in Continuum Mechanics, Vol. 378, Springer, Vienna, Austria. [CrossRef]
Agrawal, O. P. , 2008, “ A Quadratic Numerical Scheme for Fractional Optimal Control Problems,” ASME J. Dyn. Syst. Meas. Control, 130(1), p. 011010. [CrossRef]
Magin, R. L. , 2004, “ Fractional Calculus in Bioengineering,” Crit. Rev. Biomed. Eng., 32(1), pp. 1–104.
Chow, T. S. , 2005, “ Fractional Dynamics of Interfaces Between Soft-Nanoparticles and Rough Substrates,” Phys. Lett. A, 342(1–2), pp. 148–155. [CrossRef]
El-Ajou, A. , Arqub, O. A. , and Al-Smadi, M. , 2015, “ A General Form of the Generalized Taylor's Formula With Some Applications,” Appl. Math. Comput., 256(C), pp. 851–859.
El-Ajou, A. , Arqub, O. A. , Zhour, Z. A. , and Momani, S. , 2013, “ New Results on Fractional Power Series: Theories and Applications,” Entropy, 15(12), pp. 5305–5323. [CrossRef]
Rahimkhani, P. , Ordokhani, Y. , and Babolian, E. , 2016, “ Fractional-Order Bernoulli Wavelets and Their Applications,” Appl. Math. Modell., 40(17–18), pp. 8087–8107. [CrossRef]
Rahimkhani, P. , Ordokhani, Y. , and Babolian, E. , 2017, “ Numerical Solution of Fractional Pantograph Differential Equations by Using Generalized Fractional-Order Bernoulli Wavelet,” J. Comput. Appl. Math., 309, pp. 493–510. [CrossRef]
Ma, J. , Liu, J. , and Zhou, Z. , 2014, “ Convergence Analysis of Moving Finite Element Methods for Space Fractional Differential Equations,” J. Comput. Appl. Math., 255, pp. 661–670. [CrossRef]
Bhrawy, A. H. , Doha, E. H. , Baleanu, D. , and Ezz-Eldien, S. S. , 2015, “ A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of Time Fractional Diffusion-Wave Equations,” J. Comput. Phys., 293, pp. 142–156. [CrossRef]
Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations. NorthHolland Mathematics Studies, Vol. 204, Elsevier Science B.V, Amsterdam, The Netherlands.
Sabatier, J. , Agrawal, O. P. , and Machado, J. T. , 2007, Advances in Fractional Calculus, Vol. 4, Springer, Dordrecht, The Netherlands. [CrossRef]
Razminia, A. , Baleanu, D. , and Majd, V. J. , 2013, “ Conditional Optimization Problems: Fractional Order Case,” J. Optim. Theory Appl., 156(1), pp. 45–55. [CrossRef]
Ionescu, C. , Zhou, Y. , and Machado, J. T. , 2016, “ Special Issue: Advances in Fractional Dynamics and Control,” J. Vib. Control, 22(8), pp. 1969–1971. [CrossRef]
Valério, D. , Trujillo, J. J. , Rivero, M. , Machado, J. T. , and Baleanu, D. , 2013, “ Fractional Calculus: A Survey of Useful Formulas,” Eur. Phys. J. Spec. Top., 222(8), pp. 1827–1846. [CrossRef]
Alaviyan Shahri, E. S. , Alfi, A. , and Tenreiro Machado, J. A. , 2017, “ Robust Stability and Stabilization of Uncertain Fractional Order Systems Subject to Input Saturation,” J. Vib. Control, epub.
Lu, J. G. , and Chen, Y. , 2013, “ Stability and Stabilization of Fractional-Order Linear Systems With Convex Polytopic Uncertainties,” Fractional Calculus Appl. Anal., 16(1), pp. 142–157.
Agrawal, O. P. , 2004, “ A General Formulation and Solution Scheme for Fractional Optimal Control Problems,” Nonlinear Dyn., 38(1–4), pp. 323–337. [CrossRef]
Rakhshan, S. A. , Kamyad, A. V. , and Effati, S. , 2016, “ An Efficient Method to Solve a Fractional Differential Equation by Using Linear Programming and Its Application to an Optimal Control Problem,” J. Vib. Control, 22(8), pp. 2120–2134. [CrossRef]
Rakhshan, S. A. , Effati, S. , and Vahidian Kamyad, A. , 2016, “ Solving a Class of Fractional Optimal Control Problems by the Hamilton-Jacobi-Bellman Equation,” J. Vib. Control, 24(9), pp. 1741–1756.
Rakhshan, S. A. , and Effati, S. , 2018, “ The Laplace-Collocation Method for Solving Fractional Differential Equations and a Class of Fractional Optimal Control Problems,” Optim. Control Appl. Methods, 39(2), pp. 1110–1129. [CrossRef]
Wang, Q. , Chen, F. , and Huang, F. , 2016, “ Maximum Principle for Optimal Control Problem of Stochastic Delay Differential Equations Driven by Fractional Brownian Motions,” Optim. Control Appl. Methods, 37(1), pp. 90–107. [CrossRef]
Witayakiattilerd, W. , 2013, “ Optimal Regulation of Impulsive Fractional Differential Equation With Delay and Application to Nonlinear Fractional Heat Equation,” J. Math. Res., 5(2), p. 94. [CrossRef]
Kharatishvili, G. L. , 1961, “ The Maximum Principle in the Theory of Optimal Processes Involving Delay,” Dokl. Akad. Nauk USSR, 136(1), pp. 39–42.
Wang, X. T. , 2007, “ Numerical Solutions of Optimal Control for Time Delay Systems by Hybrid of Block-Pulse Functions and Legendre Polynomials,” Appl. Math. Comput., 184(2), pp. 849–856.
Khellat, F. , 2009, “ Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets,” J. Optim. Theory Appl., 143(1), pp. 107–121. [CrossRef]
Kolmanovsky, I. V. , and Maizenberg, T. L. , 2001, “ Optimal Control of Continuous-Time Linear Systems With a Time-Varying, Random Delay,” Syst. Control Lett., 44(2), pp. 119–126. [CrossRef]
Mirhosseini-Alizamini, S. M. , Effati, S. , and Heydari, A. , 2015, “ An Iterative Method for Suboptimal Control of Linear Time-Delayed Systems,” Syst. Control Lett., 82, pp. 40–50. [CrossRef]
Jarad, F. , Abdeljawad, T. , and Baleanu, D. , 2010, “ Fractional Variational Optimal Control Problems With Delayed Arguments,” Nonlinear Dyn., 62(3), pp. 609–614. [CrossRef]
Safaie, E. , Farahi, M. H. , and Ardehaie, M. F. , 2015, “ An Approximate Method for Numerically Solving Multi-Dimensional Delay Fractional Optimal Control Problems by Bernstein Polynomials,” Comput. Appl. Math., 34(3), pp. 831–846. [CrossRef]
Hosseinpour, S. , and Nazemi, A. , 2016, “ A Collocation Method Via Block-Pulse Functions for Solving Delay Fractional Optimal Control Problems,” IMA J. Math. Control Inf., 34(4), pp. 1215–1237.
Love, E. R. , and Young, L. C. , 1938, “ On Fractional Integration by Parts,” Proc. London Math. Soc., 2(1), pp. 1–35. [CrossRef]
Lalwani, C. S. , and Desai, R. C. , 1973, “ The Maximum Principle for Systems With Time-Delay,” Int. J. Control, 18(2), pp. 301–304. [CrossRef]
Marzban, H. R. , and Razzaghi, M. , 2004, “ Optimal Control of Linear Delay Systems Via Hybrid of Block-Pulse and Legendre Polynomials,” J. Franklin Inst., 341(3), pp. 279–293. [CrossRef]
Khellat, F. , and Vasegh, N. , 2011, “ Suboptimal Control of Linear Systems With Delays in State and Input by Orthonormal Basis,” Int. J. Comput. Math., 88(4), pp. 781–794. [CrossRef]
Haddadi, N. , Ordokhani, Y. , and Razzaghi, M. , 2012, “ Optimal Control of Delay Systems by Using a Hybrid Functions Approximation,” J. Optim. Theory Appl., 153(2), pp. 338–356. [CrossRef]
Nazemi, A. , and Shabani, M. M. , 2014, “ Numerical Solution of the Time-Delayed Optimal Control Problems With Hybrid Functions,” IMA J. Math. Control Inf., 32(3), pp. 623–638. [CrossRef]
Nazemi, A. , and Mansoori, M. , 2016, “ Solving Optimal Control Problems of the Time-Delayed Systems by Haar Wavelet,” J. Vib. Control, 22(11), pp. 2657–2670. [CrossRef]


Grahic Jump Location
Fig. 1

Approximation of state and control variable of Example 4.1 with different α: (a) state variable x(t) and (b) control variable u(t)

Grahic Jump Location
Fig. 2

Approximation of state and control variable of Example 4.2 with different α: (a) state variable x(t) and (b) control variable u(t)

Grahic Jump Location
Fig. 3

Approximation of state variable of Example 4.3 with α = 1: (a) state variable x1(t) and (b) state variable x2(t)

Grahic Jump Location
Fig. 4

Absolute error of control variable of Example 4.3 with α = 1

Grahic Jump Location
Fig. 5

Approximation of state variable of Example 4.4 with α = 1: (a) state variable x1(t) and (b) state variable x2(t)

Grahic Jump Location
Fig. 6

Absolute error of control variable of Example 4.4 with α = 1



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In