Skip to main content
SpringerLink
Log in

Optimality Conditions for Systems with Distributed Parameters Based on the Dubovitskii–Milyutin Theorem with Incomplete Information About the Initial Conditions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we consider an optimal control problem for a system described by a parabolic linear partial differential equation with the Neumann boundary condition. We impose some constraints on the control. The performance functional has the integral form. The control time T is fixed. The initial condition is not given by a known function but belongs to a certain set (incomplete information about the initial state). To obtain optimality conditions for the Neumann problem, a generalization of the Dubovitskii–Milyutin theorem was applied. The problem formulated in this paper describes the process of optimal heating for which we do not have exact information about the initial temperature of the heating object. We also present an example in which admissible controls and one of the initial conditions are given by means of the norm constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. M. Bahaa, “Quadratic Pareto optimal control of parabolic equation with state-control constraints and an infinite number of variables,” IMA J. Math. Control Inform., 20, 167–178 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. M. Bahaa, “Time-optimal control problem for parabolic equations with control constraints and infinite number of variables,” IMA J. Math. Control Inform., 22, 364–375 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. M. Bahaa, “Optimal control for cooperative parabolic systems governed by Schrödinger operator with control constraints,” IMA J. Math. Control Inform., 24, 1–12 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. M. Bahaa, “Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints,” IMA J. Math. Control Inform., 25, 37–48 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. M. Bahaa, “Boundary control problem of infinite order distributed hyperbolic systems involving time lags,” Intel. Control Automat., 3, No. 3, 211–221 (2012).

    Article  Google Scholar 

  6. G. M. Bahaa, “Optimality conditions for infinite order distributed parabolic systems with multiple time delays given in integral form,” J. Appl. Math., 2012, 1–25 (2012).

    MathSciNet  MATH  Google Scholar 

  7. G. M. Bahaa, “Fractional optimal control problem for variational inequalities with control constraints,” IMA J. Math. Control Inform., 33, No. 3, 1–16 (2016).

    MathSciNet  Google Scholar 

  8. G. M. Bahaa, “Fractional optimal control problem for differential system with control constraints,” Filomat., 30, No. 8, 2177–2189 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. G. M. Bahaa, “Fractional optimal control problem for infinite order system with control constraints,” Adv. Differ. Equations., 250 (2016).

  10. G. M. Bahaa, “Fractional optimal control problem for differential system with delay argument,” Adv. Differ. Equations., 69 (2017).

  11. G. M. Bahaa and W. Kotarski, “Optimality conditions for n × n infinite order parabolic coupled systems with control constraints and general performance index,” IMA J. Math. Control Inform., 25, 49–57 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  12. G. M. Bahaa and W. Kotarski, “Time-optimal control of infinite order distributed parabolic systems involving multiple time-varying lags,” Numer. Funct. Anal. Optim., No. 9, 1066–1088 (2016).

  13. G. M. Bahaa and M. M. Tharwat, “Optimal control problem for infinite variables hyperbolic systems with time lags,” Arch. Control Sci., 21, No. 4, 373–393 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. M. Bahaa and M. M. Tharwat, “Time-optimal control of infinite order parabolic system with time lags given in integral form,” J. Inform. Optim. Sci., 33, No. 2-3, 233–258 (2012).

    MathSciNet  MATH  Google Scholar 

  15. G. M. Bahaa and M. M. Tharwat, “Optimal boundary control for infinite variables parabolic systems with time lags given in integral form,” Iran. J. Sci. Technol., A3, 277–291 (2012).

    MathSciNet  MATH  Google Scholar 

  16. Yu. A. Dubinskii, “Sobolev spaces of infinite order and behavior of solution of some boundary-value problems with unbounded increase of the order of the equation,” Mat. Sb., 98 (140), No. 2 (10), 163–184 (1975).

  17. Yu. A. Dubinskii, “Nontriviality of Sobolev spaces of infinite order for a full Euclidean space and a torus,” Mat. Sb., 100 (142), No. 3 (7), 436–446 (1976).

  18. A. Ya. Dubovitskii and A. A. Milyutin, “Extremal problems with constraints,” Dokl. Akad. Nauk SSSR, 149, No. 4, 759–762 (1963).

    Google Scholar 

  19. A. Ya. Dubovitskii and A. A. Milyutin, “Extremal problems with constraints,” Zh. Vychisl. Mat. Mat. Fiz., 5, No. 3, 395–453 (1965).

    MathSciNet  Google Scholar 

  20. I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer-Verlag, New York (1972).

    Book  MATH  Google Scholar 

  21. W. Kotarski, “Some problems of optimal and Pareto optimal control for distributed parameter systems,” Rep. Silesian Univ., No. 1668, 1–93 (1997).

  22. W. Kotarski, H. A. El-Saify, and G. M. Bahaa, “Optimal control of parabolic equation with an infinite number of variables for nonstandard functional and time delay,” IMA J. Math. Control Inform., 19, No. 4, 461–476 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Kowalewski and W. Kotarski, “On application of Milutin–Dubovicki’s theorem to an optimal control problem for systems described by partial differential equations of hyperbolic type with time delay,” Systems Sci., 7, No. 1, 55–74 (1981).

    MathSciNet  MATH  Google Scholar 

  24. J. L. Lions, Optimal Control Of Systems Governed By Partial Differential Equations, Springer-Verlag (1971).

  25. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problem and Applications, Springer-Verlag, New York (1972).

    Book  MATH  Google Scholar 

  26. F. Tröltzsch, Optimality Conditions for Parabolic Control Problems and Applications, Teubner, Leipzig (1984).

    MATH  Google Scholar 

  27. S. Walczak, “On some control problems,” Acta Univ. Lodz. Folia Math., 1, 187–196 (1984).

    MathSciNet  MATH  Google Scholar 

  28. S. Walczak, “Some properties of cones in normed spaces and their application investigating extremal problems,” J. Optim. Theory Appl., 42, No. 2, 561–582 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  29. P. K. C. Wang, “Optimal control of parabolic systems with boundary conditions involving time delay,” SIAM J. Control., 13, 274–293 (1975).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Taibah University, Al-Madinah Al-Munwwarah, Saudi Arabia

    G. M. Bahaa

  2. Beni-Suef University, Beni-Suef, Egypt

    G. M. Bahaa

Authors
  1. G. M. Bahaa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. M. Bahaa.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 178, Optimal Control, 2020.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bahaa, G.M. Optimality Conditions for Systems with Distributed Parameters Based on the Dubovitskii–Milyutin Theorem with Incomplete Information About the Initial Conditions. J Math Sci 276, 199–215 (2023). https://doi.org/10.1007/s10958-023-06735-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06735-0

Keywords and phrases

AMS Subject Classification

Search

Navigation