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Ukrainian Mathematical Journal

, Volume 61, Issue 8, pp 1250–1263 | Cite as

Optimal control with impulsive component for systems described by implicit parabolic operator differential equations

  • L. A. Vlasenko
  • A. M. Samoilenko
Article

We study the problem of optimal control with impulsive component for systems described by abstract Sobolev-type differential equations with unbounded operator coefficients in Hilbert spaces. The operator coefficient of the time derivative may be noninvertible. The main assumption is a restriction imposed on the resolvent of the characteristic operator pencil in a certain right half plane. Applications to Sobolevtype partial differential equations are discussed.

Keywords

Bounded Linear Operator Mixed Problem Pulse Action Complex Hilbert Space Impulsive Control 
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.
    J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris (1968).zbMATHGoogle Scholar
  2. 2.
    A. V. Balakrishnan, Applied Functional Analysis [Russian translation], Nauka, Moscow (1980).zbMATHGoogle Scholar
  3. 3.
    I. Lasiecka and R.Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Abstract Parabolic Systems, Cambridge University, Cambridge (2000).Google Scholar
  4. 4.
    A. G. Butkovskii, Theory of Optimal Control of Systems with Distributed Parameters [in Russian], Nauka, Moscow (1965).Google Scholar
  5. 5.
    S. L. Sobolev, “Cauchy problem for the special case of systems that are not of the Kowalewska type,” Dokl. Akad. Nauk SSSR, 82, No. 2, 205–208 (1952).zbMATHMathSciNetGoogle Scholar
  6. 6.
    S. A. Gal’pern, “Cauchy problem for equations of the Sobolev type,” Usp. Mat. Nauk, 8, No. 5, 191–193 (1953).zbMATHGoogle Scholar
  7. 7.
    J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris (1968).zbMATHGoogle Scholar
  8. 8.
    A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).Google Scholar
  9. 9.
    S. I. Lyashko, Generalized Control of Linear Systems [in Russian], Naukova Dumka, Kiev (1998).Google Scholar
  10. 10.
    L. A. Vlasenko, A. G. Rutkas, and A. M. Samoilenko, “Problem of impulsive regulator for one dynamical system of the Sobolev type,” Ukr. Mat. Zh., 60, No. 8, 1027–1034 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D. J. Bender and A. Laub, “The linear-quadratic optimal regulator for descriptor systems,” IEEE Trans. Automatic Control, AC-32, No. 8, 672–688 (1987).CrossRefMathSciNetGoogle Scholar
  12. 12.
    S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).Google Scholar
  13. 13.
    K. Yosida, Functional Analysis [Russian translation], Mir, Moscow (1965).zbMATHGoogle Scholar
  14. 14.
    L. A. Vlasenko, Evolution Models with Implicit and Degenerate Differential Equations [in Russian], Sistemnye Tekhnologii, Dnepropetrovsk (2006).Google Scholar
  15. 15.
    A. M. Samoilenko and M. Ilolov, “On the theory of evolution equations with pulse action,” Dokl. Akad. Nauk SSSR, 316, No. 4, 822–825 (1991).MathSciNetGoogle Scholar
  16. 16.
    A. M. Samoilenko and M. Ilolov, “Inhomogeneous evolution equations with pulse action,” Ukr. Mat. Zh., 44, No. 1, 93–100 (1992).MathSciNetGoogle Scholar
  17. 17.
    A. D. Myshkis and A. M. Samoilenko, “Systems with pushes at given times,” Mat. Sb., 74, No. 2, 202–208 (1967).MathSciNetGoogle Scholar
  18. 18.
    L. A. Vlasenko, A. D. Myshkis, and A. G. Rutkas, “On one class of differential equations of parabolic type with pulse action,” Differents. Uravn., 44, No. 2, 222–231 (2008).MathSciNetGoogle Scholar
  19. 19.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).zbMATHGoogle Scholar
  20. 20.
    L. W. White, “Control problems governed by a pseudo-parabolic partial differential equation,” Trans. Amer. Math. Soc., 250, 235–246 (1979).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    V. S. Deineka and I. V. Sergienko, Optimal Control of Inhomogeneous Distributed Systems [in Russian], Naukova Dumka, Kiev (2003).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • L. A. Vlasenko
    • 1
  • A. M. Samoilenko
    • 2
  1. 1.Kharkov National UniversityKharkovUkraine
  2. 2.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

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