Home Optimal control of linear systems with fractional derivatives
Article
Licensed
Unlicensed Requires Authentication

Optimal control of linear systems with fractional derivatives

  • Ivan Matychyn EMAIL logo and Viktoriia Onyshchenko
Published/Copyright: March 13, 2018
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 1

Abstract

Problem of time-optimal control of linear systems with fractional Caputo derivatives is examined using technique of attainability sets and their support functions.

A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of set-valued maps and represents a fractional version of Pontryagin’s maximum principle.

A special emphasis is placed upon the problem of computing of the matrix Mittag-Leffler function, which plays a key role in the proposed methods. A technique for computing matrix Mittag-Leffler function using Jordan canonical form is discussed, which is implemented in the form of a MATLAB routine.

Theoretical results are supported by examples, in which the optimal control functions, in particular of the “bang-bang” type, are obtained.

MSC 2010: Primary 26A33; Secondary 34A08; 49N05
Key Words and Phrases: fractional calculus; fractional differential equations; matrix Mittag-Leffler function; optimal control

References

[1] O. Agrawal, A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, No 1–4 (2004), 323–337.10.1007/s11071-004-3764-6Search in Google Scholar

[2] O.P. Agrawal, D. Baleanu, A Hamiltonian Formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control13, No 9-10 (2007), 1269–1281.10.1177/1077546307077467Search in Google Scholar

[3] R.J. Aumann, Integrals of set valued functions. J. Math. Anal. Appl. 12, No 1 (1965), 1–12.10.1016/0022-247X(65)90049-1Search in Google Scholar

[4] R.L. Bagley, P.J. Torvik, On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51 (1984), 294–298.10.1115/1.3167615Search in Google Scholar

[5] V.I. Blagodatskikh, A.F. Filippov, Differential inclusions and optimal control. Tr. Mat. Inst. im. V.A. Steklova169 (1985), 194–252 (in Russian).Search in Google Scholar

[6] A. Chikrii, S. Eidelman, Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order. Cybern. Syst. Anal. 36, No 3 (2000), 315–338.10.1007/BF02732983Search in Google Scholar

[7] A. Chikrii, I. Matichin, Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann–Liouville, Caputo, and Miller–Ross. J. Autom. Inf. Sci. 40, No 6 (2008), 1–11.10.1615/JAutomatInfScien.v40.i6.10Search in Google Scholar

[8] A. Debbouche, D. Torres, Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fract. Calc. Appl. Anal. 18, No 1 (2015), 95–121, 10.1515/fca-2015-0007; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Search in Google Scholar

[9] K. Diethelm, J. Ford, Numerical solution of the Bagley-Torvik equation. BIT Numer. Math. 42, No 3 (2002), 490–507.10.1023/A:1021973025166Search in Google Scholar

[10] R. Gorenflo, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function Eα,β(z) and its derivative. Fract. Calc. Appl. Anal. 5, No 4 (2002), 491–518.Search in Google Scholar

[11] F.R. Gantmacher, The Theory of Matrices. AMS Chelsea Publishing, New York (1959).Search in Google Scholar

[12] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J. Numer. Anal. 53, No 3 (2015), 1350–1369.10.1137/140971191Search in Google Scholar

[13] N.J. Higham, Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008).10.1137/1.9780898717778Search in Google Scholar

[14] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[15] V. Kiryakova, A brief story about the operators of generalized fractional calculus. Fract. Calc. Appl. Anal. 11, No 2 (2008), 201–218; at http://www.math.bas.bg/complan/fcaa.Search in Google Scholar

[16] I. Matychyn, V. Onyshchenko, Time-optimal control of fractional-order linear systems. Fract. Calc. Appl. Anal. 18, No 3 (2015), 687–696; 10.1515/fca-2015-0042; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.Search in Google Scholar

[17] I.I. Matychyn, V.V. Onyshchenko, Time-optimal problem for systems with fractional dynamics. J. Autom. Inf. Sci. 48, No 8 (2016), 37–45.10.1615/JAutomatInfScien.v48.i8.40Search in Google Scholar

[18] I. Matychyn, Matrix Mittag-Leffler function. MATLAB Central File Exchange (2017); File ID: 62790.Search in Google Scholar

[19] C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, No 1 (2003), 3–49.10.1137/S00361445024180Search in Google Scholar

[20] S. Pooseh, R. Almeida, D.F.M. Torres, Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10, No 2 (2014), 363–381.10.3934/jimo.2014.10.363Search in Google Scholar

[21] B. Pshenichnyi, V. Ostapenko, differential Games. Naukova Dumka, Kiev (1992) (in Russian).Search in Google Scholar

[22] C. Tricaud, Y. Chen, Time-optimal control of systems with fractional dynamics. Int. J. Differ. Equ. 2010 (2010), Art. # 461048, 16 pp.10.1109/CDC.2009.5400637Search in Google Scholar

Received: 2017-10-12
Published Online: 2018-3-13
Published in Print: 2018-2-23

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
https://doi.org/10.1515/fca-2018-0009
Keywords for this article
fractional calculus; fractional differential equations; matrix Mittag-Leffler function; optimal control

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 16.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2018-0009/html
Scroll to top button