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Extremal Problems for Infinite Order Parabolic Systems with Time-Varying Lags

Conference paper
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Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)

Abstract

Extremal problems for infinite order parabolic systems with time-varying lags are presented. An optimal boundary control problem for infinite order parabolic systems in which time-varying lags appear in the state equations and in the boundary conditions simultaneously is solved. The time horizon is fixed. Making use of Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with quadratic performance functionals and constrained control are derived.

Keywords

Boundary control Infinite order parabolic systems Time-varying lags 

Notes

Acknowledgements

The research presented here was carried out within the research programme AGH University of Science and Technology, No. 16.16.120.773.

References

  1. 1.
    Dubinskii, J.A.: Sobolev spaces of infinite order and behavior of solution of some boundary value problems with unbounded increase of the order of the equation. Matiematiczeskii Sbornik 98, 163–184 (1975)Google Scholar
  2. 2.
    Dubinskii, J.A.: Non-trivality of Sobolev spaces of infinite order for a full Euclidean space and a Tour’s. Matiematiczeskii Sbornik 100, 436–446 (1976)Google Scholar
  3. 3.
    Dubinskii, J.A.: About one method for solving partial differential equations. Dokl. Akad. Nauk SSSR 258, 780–784 (1981)MathSciNetGoogle Scholar
  4. 4.
    Gilbert, E.S.: An iterative procedure for computing the minimum of a quadratic form on a convex set. SIAM J. Control 4(1), 61–80 (1966)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Girsanov, I. V.: Lectures on the Mathematical Theory of Extremal Problems. Publishing House of the University of Moscow, Moscow (1970). (in Russian)Google Scholar
  6. 6.
    Kowalewski, A., Miśkowicz, M.: Extremal problems for time lag parabolic systems. In: Proceedings of the 21st International Conference of Process Control (PC), 6-9 June pp. 446–451, Strbske Pleso, Slovakia (2017)Google Scholar
  7. 7.
    Kowalewski, A.: On optimal control problem for parabolic-hyperbolic system. Probl. Control Inf. Theory 15(5), 349–359 (1986)MathSciNetGoogle Scholar
  8. 8.
    Kowalewski, A.: Extremal Problems for distributed parabolic systems with boundary conditions involving time-varying lags. In: Proceedings of the 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), Miȩdzyzdroje, Poland, 28–31 August, pp. 447–452 (2017)Google Scholar
  9. 9.
    Kowalewski, A.: Optimal control of parabolic systems with time-varying lags. IMA J. Math. Control and Inf. 10(2), 113–129 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kowalewski, A.: Boundary control of distributed parabolic system with boundary condition involving a time-varying lag. Int. J. Control 48(6), 2233–2248 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kowalewski, A.: Optimal Control of Infinite Dimensional Distributed Parameter Systems with Delays. AGH University of Science and Technology Press, Cracow (2001)Google Scholar
  12. 12.
    Kowalewski, A.: Extremal problems for parabolic systems with time-varying lags. Arch. Control Sci. 28(1), 89–104 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kowalewski, A.: Extremal problems distributed parabolic systems with multiple time-varying lags. In: Proceedings of 23rd International Conference on Methods and Models in Automation and Robotics (MMAR), Miȩdzyzdroje, Poland, 27–30 August, pp. 791–796 (2018)Google Scholar
  14. 14.
    Kowalewski, A., Duda, J.: On some optimal control problem for a parabolic system with boundary condition involving a time-varying lag. IMA J. Math. Control Inf. 9(2), 131–146 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kowalewski, A., Miśkowicz, M.: Extremal problems for integral time lags parabolic systems. In: Proceedings of the 24th International Conference on Methods and Models in Automation and Robotics (MMAR), Miȩdzyzdroje, Poland, 26–29 August, pp. 7–12 (2019)Google Scholar
  16. 16.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefGoogle Scholar
  17. 17.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vols. 1 and 2. Springer, Berlin (1972)Google Scholar
  18. 18.
    Maslov, V.P. : Operators Methods, Moscow (1973). (in Russian)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Automatics and RoboticsAGH University of Science and TechnologyCracowPoland

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