Ukrainian Mathematical Journal

, Volume 63, Issue 5, pp 759–767 | Cite as

Approximate stabilization for a nonlinear parabolic boundary-value problem

  • O. V. Kapustyan
  • O. A. Kapustyan
  • A. V. Sukretna

For the problem of optimal stabilization of solutions of a nonlinear parabolic boundary-value problem with small parameter in the nonlinear term, we substantiate the form of approximate regulator on the basis of the formula of optimal synthesis of the corresponding linear-quadratic problem.


Initial Function Exact Formula Limit Equality Optimal Synthesis Optimal Stabilization 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dériveés Partielles, Dunod, Gauthier-Villars, Paris (1968).MATHGoogle Scholar
  2. 2.
    A. I. Egorov, Optimal Control over Thermal and Diffusion Processes [in Russian], Nauka, Moscow (1978).Google Scholar
  3. 3.
    J. Valero and O. V. Kapustyan, “On the connectedness and asymptotic behavior of solutions of reaction–diffusion systems,” J. Math. Anal. Appl., 323, 614–633 (2006).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    O. V. Kapustyan, O. A. Kapustyan, and A. V. Sukretna, “Approximate bounded synthesis for one weakly nonlinear boundary-value problem,” Nelin. Kolyvannya., 12, No. 3, 291–298 (2009); English translation: Nonlin. Oscillations, 12, No. 3, 297–304 (2009).MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. N. Bublik and A. I. Nevidomskii, “Synthesis of optimal concentrated control for the heat-conduction equation,” Model. Syst. Obrab. Inform., Issue 1, 78–87 (1982).Google Scholar
  6. 6.
    A.V. Sukretna, “Approximate optimal stabilization of solutions of a parabolic boundary-value problem by bounded control,” Nelin. Kolyvannya, 9, No. 2, 264–279 (2006); English translation: Nonlin. Oscillations, 9, No. 2, 257–273 (2006).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris (1969).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • O. V. Kapustyan
    • 1
  • O. A. Kapustyan
    • 1
  • A. V. Sukretna
    • 1
  1. 1.KyivUkraine

Personalised recommendations