Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 39. no. 2. pp. 62–77. ISSN 2079-6641

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MSC 35G15; 35L35

Research Article

On a control problem for the subdiffusion equation with a fractional derivative in the sense of Caputo

Yu. E. Fayziev

National University of Uzbekistan Uzbekistan, 100174, Tashkent city, university campus, Republic of Uzbekistan.


In the rectangle for a differential equation of fractional order in the sense of Caputo, we study the control problem with the help of a source function. In other words, the task is to find the source function f(x;y) in such a way that, as a result, at the time t = Θ the temperature of the object under study should be distributed as a given function Ψ(x;y). Sufficient conditions are found for the function Ψ(x;y), which ensure both the existence and uniqueness of the solution to the control problem.

Key words: fractional derivatives in the sense of Caputo, heat conduction equations, control problem.

DOI: 10.26117/2079-6641-2022-39-2-62-77

Original article submitted: 20.06.2022

Revision submitted: 12.08.2022

For citation. Fayziev Yu. E. On one control problem for the equationOn a control problem for the subdiffusion equation with a fractional derivative in the sense of Caputo. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 62-77. DOI: 10.26117/2079-6641-2022-39-2-62-77

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Fayziev Yu. E., 2022


  1. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006, p. 540.
  2. Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Fractional Partial Differential Equations], Moscow, Nauka, 2005 (In Russian).
  3. Lions J. L. Control Optimal de Systems Governess par des Equations aux Derivees Partielles, Paris, Dunod, 1968.
  4. Il’in V. A. Boundary Control of Oscillations at One Endpoint with the Other Endpoint Fixed in Terms of a Finite-Energy Generalized Solution of the Wave Equation, Differential Equations, 2000, vol. 36, pp. 1832–1849.
  5. Il’in V. A., Moiseev E. I. Optimal boundary control of a string by an elastic force applied at one end, the other end being free, Differential Equations, 2005, vol. 41, pp. 110–120.
  6. Il’in V. A., Moiseev E. I. Optimization of a boundary control by a displacement at one end of a string based on minimization of the integral of the modulus of the derivative of the displacement raised to an arbitrary power p >-1, Doklady Mathematics, 2006, vol. 74, no 3, pp. 878-882.
  7. Fattorini H. O. Time and norm optimal control for linear parabolic equations: necessary and sufficient conditions, Control and Estimation of Distributed Parameter Systems. International Series of Numerical Mathematics, 2002, vol. 13, pp. 151–168.
  8. Barbu V., Rascanu A., Tessitore G. Carleman estimates and controllability of linear stochastic heat equations, Appl. Math. Optim., 2003, vol. 47, pp. 97–120.
  9. Alimov Sh. A. O zadachi bystrodeystviya v upravlenii protsessom teploobmena, Uzbek Mathematical Journal, 2005, no. 4. pp. 13–21 (In Russian.).
  10. Alimov Sh. A. Ob odnoy zadache upravleniya protsessom teploobmena, Doklady akademii nauk, 2008, vol. 421, no 4, pp. 583–585 (In Russian.).
  11. Alimov Sh. A., Albeverio S. On a Time-Optimal Control Problem Associated with the Heat Exchange Process, Appl. Math. Optim., 2008, no. 572, pp. 58–68.
  12. Alimov Sh. A. On a control problem associated with the heat transfer process, Eurasian mathematical journal, 2010, vol. 1. no. 2, pp. 17–30.
  13. Alimov Sh. A. On the null-controllability of the heat exchange process, Eurasian mathematical journal, 2011, vol. 2, no. 3, pp. 5–19.
  14. Fayziyev Yu. E., Khalilova N. Ob odnoy zadache upravleniya protsessom teploprovodnosti, Vestnik NUUz, 2016, no 2/1, pp. 49–54 (In Russian).
  15. Fayziyev YU. E., Kuchkarov A. F., Nosirova D. E. Ob odnoy zadache upravleniya protsessom teploprovodnosti v pryamougol’nike, Vestnik NUUz, 2017, no. 2/2, pp. 239–244 (In Russian).
  16. Fayziev Yu. E. On the control of heat conduction, IIUM Engineering Journal, 2018, vol. 19, no. 1, pp. 168–177.
  17. Liu Y., Li Z., Yamamoto M. Inverse problems of determining sources of the fractional partial differential equations, Handbook of Fractional Calculus with Applications, 2019, vol. 2, pp. 411–430.
  18. Ashurov R., FayzievYu. On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations, Fractal and Fractional, 2022, vol. 6, no. 41, pp. 168–177.
  19. Ashyralyev A., Urun M. Time-dependent source identification Schrodinger type problem, International Journal of Applied Mathematics, 2021, vol. 34. no. 2. pp. 297–310.
  20. Niu P., Helin T., Zhang Z. An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 2020, vol. 36, no. 4, 045002.
  21. Slodichka M. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation, Frac. Cal. and Appl. Anal. 2020, vol. 23, no. 6, pp. 1703-1711.
  22. Zhang Y., Xu X. Inverse scource problem for a fractional differential equations, Inverse Probems, 2011, vol. 27. no. 3, pp. 31–42.
  23. Kirane M., Malik A. S. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Applied Mathematics and Computation, 2011, vol. 218, pp. 163–170.
  24. Kirane M., Samet B., Torebek B. T. Determination of an unknown source term and the temperature distribution for the subdiffusion equation at the initial and final data, Electronic Journal of Differential Equations, 2017, vol. 217, pp. 1–13.
  25. Nguyen H. T., Le D. L., Nguyen V. T. Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling, 2016, vol. 40, pp. 8244–8264.
  26. Li Z., Liu Y., Yamamoto M. Initial-boundary value problem for multi-term timefractional diffusion equation with positive constant coefficients, Applied Mathematica and Computation, 2015, vol. 257, pp. 381–397.
  27. Rundell W., Zhang Z. Recovering an unknown source in a fractional diffusion problem, Journal of Computational Physics, 2018, vol. 386, pp. 299–314.
  28. Malik S. A., Aziz S. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions, Computers and Mathematics with applications, 2017, vol. 3, pp. 7–19.
  29. Ruzhansky M., Tokmagambetov N., Torebek B. T. Inverse source problems for positive operators, J. Inverse Ill-Possed Probl, 2019, vol. 27, pp. 891–911.
  30. Ashurov R. R. Muhiddinova O. Inverse problem of determining the heat source density for the subdiffusion equation, Differential equations, 2020, vol. 56, no. 12, pp. 1550–1563.
  31. Ashurov R., Fayziev Yu. On construction of solutions of linear fractional differential equations with constant coefficients and the fractional derivatives, Uzb. Math. Journ., 2017, no. 3, pp. 3–21.
  32. Shuang Zh., Saima R., Asia R., Khadija K., Abdullah M. Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, AIMS Mathematics, 2021, vol. 6, no. 11, pp. 12114–12132.
  33. Ashurov R., Fayziev Yu. Inverse problem for determining the order of the fractional derivative in the wave equation, Mathematical Notes, 2021, vol. 110, no. 6. pp. 842–852.
  34. Kirane M., Salman A. M. Mohammed A. Al-Gwaiz An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci. 2013, vol. 36, no. 9, pp. 1056-1069.
  35. Ashurov R., Fayziev Yu. Determination of fractional order and source term in a fractional subdiffusion equation, Eurasian Mathematical Journal, 2022, vol. 13, no. 1, pp. 19–31.
  36. Ashurov R., Fayziev Yu. Uniqueness and existence for inverse problem of determining an order of time-fractional derivative of subdiffusion equation, Lobachevskii journal of mathemtics, 2021, vol. 42, no. 3, pp. 508–516.

Fayziev Yusuf Ergashevich – PhD (Phys. & Math.), Associate Professor, Faculty of Physics and Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan, ORCID 0000-0002-8361-2525.