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Ukrainian Mathematical Journal

, Volume 67, Issue 7, pp 1091–1102 | Cite as

Optimal Control over Moving Sources in the Heat Equation

  • R. A. Teimurov
Article
  • 41 Downloads

We study the problem of optimal control over the processes described by the heat equation and a system of ordinary differential equations. For the problem of optimal control, we prove the existence and uniqueness of solutions, establish sufficient conditions for the Fréchet differentiability of the purpose functional, deduce the expression for its gradient, and obtain necessary conditions of optimality in the form of an integral maximum principle.

Keywords

Heat Equation Admissible Control Reflexive Banach Space Pontryagin Function Unique Generalize Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. G. Butkovskii, Methods of Control over Systems with Distributed Parameters [in Russian], Nauka, Moscow (1965).Google Scholar
  2. 2.
    A. G. Butkovskii and L. M. Pustyl’nikov, Theory of Moving Control over Systems with Distributed Parameters [in Russian], Nauka, Moscow (1980).Google Scholar
  3. 3.
    J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris (1968).zbMATHGoogle Scholar
  4. 4.
    S. I. Lyashko, Generalized Control over Linear Systems [in Russian], Naukova Dumka, Kiev (1998).Google Scholar
  5. 5.
    J. Droniou and J.-P. Raymond, “Optimal pointwise control of semilinear parabolic equations,” Nonlin. Anal., 39, 135–156 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    D. Leykekhman and B. Vexler, “Optimal a priori error estimates of parabolic optimal control problems with pointwise control,” SIAM J. Numer. Anal., 51, 2797–2821 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    D. Meidner, R. Rannacher, and B. Vexler, “A priori error estimates for finite-element discretizations of parabolic optimization problems with poinwise state constants in time,” SIAM J. Control Optim., 49, 1961–1997 (2011).CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    W. Gong, M. Hinze, and Z. Zhou, “A priori error estimates for finite-element approximation of parabolic optimal control problems with poinwise control,” SIAM J. Control Optim., 52, 97–119 (2014).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    K. Kunisch, K. Pieper, and B. Vexler, “Measure valued directional sparsity for parabolic optimal control problems,” SIAM J. Control Optim., 52, 3078–3108 (2014).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    R. A. Teimurov, “On the problem of optimal control over moving sources for heat equations,” Izv. Vyssh. Uchebn. Zaved., Severo-Kavkaz. Region, Ser. Estestven. Nauk., No. 4, 17–20 (2012).Google Scholar
  11. 11.
    R. A. Teimurov, “On the control problem by moving sources for systems with distributed parameters,” Vestn. Tomsk. Gos. Univ., Ser. Mat. Mekh., No. 1(21), 24–33 (2013).Google Scholar
  12. 12.
    F. P. Vasil’ev, Methods for the Solution of Extremal Problems [in Russian], Nauka, Moscow (1981).Google Scholar
  13. 13.
    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1976).Google Scholar
  14. 14.
    J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications [Russian translation], Mir, Moscow (1971).Google Scholar
  15. 15.
    A. N. Tikhonov and V. Ya. Arsenin, Methods for the Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1974).Google Scholar
  16. 16.
    O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  17. 17.
    V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1983).Google Scholar
  18. 18.
    K. Yosida, Functional Analysis, Springer, Berlin (1965).CrossRefzbMATHGoogle Scholar
  19. 19.
    M. Goebel, “On existence of optimal control,” Math. Nachr., 93, 67–73 (1979).CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • R. A. Teimurov
    • 1
  1. 1.Institute of Mathematics and MechanicsAzerbaijan National Academy of SciencesBakuAzerbaijan

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