Skip to main content
SpringerLink
Log in

An optimal control problem for a nonlinear elliptic equation with a phase constraint and state variation

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study an optimal control problem for a system described by a nonlinear elliptic equation with a state constraint in the form of an inclusion. We prove the solvability of the problem under consideration and by varying the state of the system obtain necessary optimality conditions.

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. A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications (Nauchnaya Kniga, Novosibirsk, 1999) [in Russian].

    MATH  Google Scholar 

  2. J.-L. Lions, Control of Distributed Singular Systems (Dunod-Gauthier-Villars, Paris, 1985; Nauka, Moscow, 1987).

    Google Scholar 

  3. G. F. Kuliev and K. K. Gasanov, “Necessary Conditions of Optimality for Systems with Distributed Parameters and Control in the Coefficients of High Derivatives,” Differents. Uravneniya 18(6), 1028–1036 (1982).

    MathSciNet  MATH  Google Scholar 

  4. A. S. Matveev and V. A. Yakubovich, Abstract Theory of Optimal Control (St. Peterburg Univ., St. Petersburg, 1994) [in Russian].

    Google Scholar 

  5. P. Neittaanmäki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms, and Applications (Marcel Dekker, New York, 1994).

    MATH  Google Scholar 

  6. E. Casas, F. Tröltzsch, and A. Unger, “Second Order Sufficient Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations,” SIAM J. Contr. Optim. 38(5), 1369–1381 (2000).

    Article  MATH  Google Scholar 

  7. V. Barbu and G. Wang, “Internal Stabilization of Semilinear Parabolic Systems,” J. Math. Anal. Appl. 285(2), 387–407 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Griesse and S. Volkwein, “A Primal-Dual Active Set Strategy for Optimal Boundary Control of a Nonlinear Reaction-Diffusion System,” SIAM J. Contr. Optim. 44(2), 467–494 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. R. Henríquez and E. Hernández, “Stabilization of Linear Distributed Control Systems with Unbounded Delay,” J. Math. Anal. Appl. 307(1), 321–338 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Rösch and F. Tröltzsch, “On Regularity of Solutions and Lagrange Multipliers of Optimal Control Problems for Semilinear Equations with Mixed Pointwise Control-State Constraints,” SIAM J. Contr. Optim. 46(3), 1098–1115 (2007).

    Article  MATH  Google Scholar 

  11. S. Ya. Serovaiskii, “Optimal Control of a Singular Evolution Equation with a Nonsmooth Operator and Fixed Terminal State,” Differents. Uravneniya 43(2), 251–258 (2007).

    MathSciNet  Google Scholar 

  12. H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie-Verlag, Berlin, 1974; Mir, Moscow, 1978).

    MATH  Google Scholar 

  13. Functional Analysis, Ed. by S. G. Krein (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  14. J.-L. Lions, Optimal Control of Systems Described by Linear Partial Differential Equations (C. R. of the Academy of Sciences, Paris, 1966; Mir, Moscow, 1972).

    Google Scholar 

  15. R. Glowinski, J.-L. Lions, and R. Trëmoliëres, Numerical Analysis of Variational Inequalities (North-Holland, Amsterdam, 1981; Mir, Moscow, 1979).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Al-Farabi Kazakh National University, ul. Al-Farabi 71, Almaty, 050078, Republic of Kazakhstan

    S. Ya. Serovaiskii

Authors
  1. S. Ya. Serovaiskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Ya. Serovaiskii.

Additional information

Original Russian Text © S.Ya. Serovaiskii, 2013, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2013, No. 9, pp. 81–86.

Submitted by V.A. Srochko

About this article

Cite this article

Serovaiskii, S.Y. An optimal control problem for a nonlinear elliptic equation with a phase constraint and state variation. Russ Math. 57, 67–70 (2013). https://doi.org/10.3103/S1066369X13090119

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X13090119

Keywords and phrases

Search

Navigation