Automation and Remote Control

, Volume 70, Issue 4, pp 577–588 | Cite as

N.N. Krasovskii’s extremal shift method and problems of boundary control

  • Yu. S. Osipov
  • A. V. Kryazhimskii
  • V. I. Maksimov
Systems with Distributed Parameters


For the boundary-controlled dynamic system obeying a parabolic differential equation with the Neumann boundary condition, the problems of following the reference motion, following the reference control, and guaranteed control (at domination of the controller resource) were solved on the basis of the N.N. Krasovskii method of extremal shift from the theory of positional differential games.

PACS number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Nauka, 1974.Google Scholar
  2. 2.
    Osipov, Yu.S. and Kryazhimskii, A.V., Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, London: Gordon and Breach, 1995.MATHGoogle Scholar
  3. 3.
    Osipov, Yu.S., Pandolfi, L., and Maksimov, V.I., Problems of Dynamical Reconstruction and Robust Boundary Control: The Case of Dirichlet Boundary Conditions, Collected Papers Int. Conf. “Distributed Systems: Optimization and Applications in Economics and Environmental Sciences” (DSO’2000), Ekaterinburg, 2000, pp. 42–45.Google Scholar
  4. 4.
    Osipov, Yu.S., Differential Games in Systems with Aftereffect, Dokl. Akad. Nauk SSSR, 1971, vol. 196, no. 4, pp. 761–768.MathSciNetGoogle Scholar
  5. 5.
    Osipov, Yu.S., Positional Modeling in Parabolic Systems, Prikl. Mat. Mekh., 1977, vol. 42, no. 2, pp. 341–376.Google Scholar
  6. 6.
    Osipov, Yu.S., Pandolfi, L., and Maksimov, V.I., Problem of Robust Boundary Control: The case of Dirichlet Boundary Conditions, Dokl. Ross. Akad. Nauk, 2000, vol. 374, no. 3, pp. 310–312.MathSciNetGoogle Scholar
  7. 7.
    Kryazhimskii, A.V. and Osipov, Yu.S., On Dynamic Solution of the Operator Equations, Dokl. Akad. Nauk SSSR, 1983, vol. 269, no. 3, pp. 552–556.MathSciNetGoogle Scholar
  8. 8.
    Kryazhimskii, A.V. and Osipov, Yu.S., On Modeling Control in the Dynamic System, Izv. Ross. Akad. Nauk, Tekh. Kibern., 1983, no. 2, pp. 51–60.Google Scholar
  9. 9.
    Osipov, Yu.S., Pandolfi, L., and Maksimov, V.I., Problems of Dynamic Reconstruction and Robust Boundary Control: The Case of Dirichlet Boundary Conditions, J. Inverse Ill-Posed Probl., 2001, no. 9(2), pp. 149–162.Google Scholar
  10. 10.
    Kryazhimskii, A.V., Maksimov, V.I., and Osipov, Yu.S., Dynamic Inverse Problems for the Parabolic Systems, Diff. Uravn., 2000, vol. 37, no. 5, pp. 579–597.MathSciNetGoogle Scholar
  11. 11.
    Kryazhimskii, A.V. and Osipov, Yu.S., Extremal Problems with Separable Graphs, Kibern. Sist. Anal., 2002, no. 2, pp. 32–55.Google Scholar
  12. 12.
    Kryazhimskii, A.V. and Osipov, Yu.S., Method of Extremal Shift and the Problems of Optimization, Proc. Inst. Mat. Mech., Ural Branch, Russ. Acad. Sci., Ekaterinburg, 2004, vol. 10, no. 2, pp. 83–105.MathSciNetGoogle Scholar
  13. 13.
    Lasiecka, I. and Triggiani, R., Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lecture Notes Control Inform. Sci., Berlin: Springer, 1991, vol. 164.CrossRefGoogle Scholar
  14. 14.
    Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaries, Paris: Dunod Gauthier-Villars, 1969. Translated under the title Nekotorye metody resheniya nelineinykh kraevykh zadach, Moscow: Mir, 1972.Google Scholar
  15. 15.
    Tikhonov, A.N. and Arsenin, V.Ya., Metody resheniya nekorrektnykh zadach (Methods to Solve Ill-posed Problems), Moscow: Nauka, 1978.Google Scholar
  16. 16.
    Kryazhimskii, A.V., Maksimov, V.I., and Osipov, Yu.S., On the Procedure of Solution of the Control Problem with Phase Constraints, Zh. Vychisl. Mat. Mat. Fiz., 1998, vol. 38, no. 9, pp. 1484–1489.MathSciNetGoogle Scholar
  17. 17.
    Barbu, V., Boundary Control Problems with Convex Cost Criterion, SIAM J. Control Optim., 1980, vol. 18, no. 2, pp. 227–243.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lasiecka, I., Boundary Control of Parabolic Systems: Regularity of Optimal Solutions, Appl. Math. Optim., 1978, vol. 4, no. 4, pp. 301–328.MATHMathSciNetGoogle Scholar
  19. 19.
    Tröltzsch, F., On Convergence of Semidiscrete Ritz-Galerkin Schemes Applied to the Boundary Control of Parabolic Equations with Non-linear Boundary Condition, ZAMM, 1992, vol. 72, no. 7, pp. 291–301.MATHCrossRefGoogle Scholar
  20. 20.
    Lasiecka, I., Unified Theory for Abstract Parabolic Boundary Problems—A Semigroup Approach, Appl. Math. Optim., 1980, vol. 6, no. 4, pp. 287–334.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ivanov, V.K., Vasin, V.V., and Tanana, V.P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya (Theory of Ill-posed Linear Problems and Its Applications), Moscow: Nauka, 1978.Google Scholar
  22. 22.
    Maksimov, V.I. and Pandolfi, L., Dynamical Reconstruction of Inputs for Contraction Semigroup Systems: The Boundary Input Case, J. Optim. Theory Appl., 1999, vol. 103, no. 2, pp. 399–410.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • Yu. S. Osipov
    • 1
  • A. V. Kryazhimskii
    • 2
  • V. I. Maksimov
    • 3
  1. 1.Presidium of the Russian Academy of SciencesMoscowRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Institute of Mathematics and Mechanics, Ural BranchRussian Academy of SciencesYekaterinburgRussia

Personalised recommendations