Automation and Remote Control

, Volume 79, Issue 5, pp 884–896 | Cite as

Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

  • V. A. Kubyshkin
  • S. S. PostnovEmail author
Intellectual Control Systems, Data Analysis


We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.


optimal control diffusion equation Caputo’s fractional derivative the problem of moments 


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  1. 1.
    Uchaikin, V.V., Metod drobnykh proizvodnykh (Method of Fractional Derivatives), Ul’yanovsk: Artishok, 2008.Google Scholar
  2. 2.
    Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.zbMATHGoogle Scholar
  3. 3.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., et al., Fractional-Order Systems and Controls: Fundamentals and Applications, London: Springer-Verlag, 2010.CrossRefzbMATHGoogle Scholar
  4. 4.
    Caponetto, R., Dongola, G., Fortuna, L., and Petras, I., Fractional Order Systems. Modeling and Control Applications, Singapore: World Scientific, 2010.CrossRefGoogle Scholar
  5. 5.
    Tarasov, V.E., Fractional Dynamics, Berlin: Springer, 2010.CrossRefzbMATHGoogle Scholar
  6. 6.
    Sandev, T., Metzler, R., and Tomovski, Z., Fractional Diffusion Equation with a Generalized Riemann–Liouville Time Fractional Derivative, J. Phys. A: Math. Theor., 2011, vol. 44, p. 255203.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Tomovski, Z. and Sandev, T., Exact Solutions for Fractional Diffusion Equation in a Bounded Domain with Different Boundary Conditions, Nonlin. Dyn., 2013, vol. 71, pp. 671–683.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Podlubny, I., Fractional Differential Equations, San Diego: Academic, 1999.zbMATHGoogle Scholar
  9. 9.
    Agrawal, O.P., A General Formulation and Solution Scheme for Fractional Optimal Control Problems, Nonlin. Dyn., 2004, vol. 38, pp. 323–337.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frederico, G.S.F. and Torres, D.F.M., Fractional Optimal Control in the Sense of Caputo and the Fractional Noether’s Theorem, Int. Math. Forum, 2008, vol. 3, no. 10, pp. 479–493.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Mophou, G.M., Optimal Control of Fractional Diffusion Equation, Comput. Math. Appl., 2011, vol. 61, pp. 68–78.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tang, Q. and Ma, Q., Variational Formulation and Optimal Control of Fractional Diffusion Equations with Caputo Derivatives, Adv. Diff. Eq., 2015, vol. 283, DOI 10.1186/s13662-015-0593-5.Google Scholar
  13. 13.
    Zhou, Z. and Gong, W., Finite Element Approximation of Optimal Control Problems Governed by Time Fractional Diffusion Equation, Comput. Math. Appl., 2016, vol. 71, pp. 301–318.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami (Optimal Control Theory for Systems with Distributed Parameters), Moscow: Nauka, 1965.Google Scholar
  15. 15.
    Kubyshkin, V.A. and Postnov, S.S., Optimal Control Problem for a Linear Stationary Fractional Order System in the Form of a Problem of Moments: Problem Setting and a Study, Autom. Remote Control, 2014, vol. 75, no. 5, pp. 805–817.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kubyshkin, V.A. and Postnov, S.S., The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters, Discontinuity, Nonlin. Complexity, 2015, vol. 4 (4), pp. 429–443.CrossRefzbMATHGoogle Scholar
  17. 17.
    Kubyshkin, V.A. and Postnov, S.S., The Optimal Control Problem for Linear Distributed Systems of Fractional Order, Vestn. Ross. Univ. Druzhby Narodov, Ser. Mat., Fiz., Informatika, 2014, no. 2, pp. 381–385.zbMATHGoogle Scholar
  18. 18.
    Egorov, A.I., Optimal’noe upravlenie teplovymi i diffuzionnymi protsessami (Optimal Control for Heat and Diffusion Processes), Moscow: Nauka, 1978.Google Scholar
  19. 19.
    Butkovskii, A.G., Malyi, S.A., and Andreev, Yu.N., Upravlenie nagrevom metalla (Control over the Heating of Metal), Moscow: Metallurgiya, 1981.Google Scholar
  20. 20.
    Egorov, Yu.V., On Some Problems in Optimal Control Theory, Dokl. Akad. Nauk SSSR, 1962, vol. 145, no. 4, pp. 720–723.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Agrawal, O.P., Fractional Variational Calculus in Terms of Riesz Fractional Derivatives, J. Phys. A: Math. Theor., 2007, vol. 40, pp. 6287–6303.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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