Skip to main content
SpringerLink
Log in

Necessary Optimality Conditions for Singular Controls in Stochastic Goursat–Darboux Systems

  • STOCHASTIC SYSTEMS
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We consider an optimal control problem for a stochastic system whose dynamics is described by a second-order hyperbolic stochastic partial differential equation with Goursat boundary conditions. A stochastic analog of Pontryagin’s maximum principle is obtained, and singularities in the sense of the control maximum principle are analyzed for optimality.

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. Egorov, A.I., Concerning optimum control of processes in some systems with distributed parameters, Autom. Remote Control, 1964, vol. 25, no. 5, pp. 613–623.

    Google Scholar 

  2. Srochko, V.A., Conditions for optimality for a class of distributed parameter systems, Sib. Math. J., 1976, vol. 17, no. 5, pp. 819–825.

    Article  Google Scholar 

  3. Ashchepkov, L.T., Vasil’ev, O.V., and Kovalenok, I.L., Strengthened optimality condition for singular controls in the Goursat–Darboux system, Differ. Equations, 1980, no. 6, pp. 1054–1059.

  4. Novozhenov, M.M, Sumin, V.I., and Sumin, M.I., Metody optimal’nogo upravleniya sistemami matematicheskoi fiziki (Methods for Optimal Control of Systems of Mathematical Physics), Gorky: Izd. Gor’kovsk. Gos. Univ., 1986.

    Google Scholar 

  5. Mansimov, K.B., Multiple point necessary conditions for optimality of quasi-singular controls, Autom. Remote Control, 1982, vol. 43, no. 10, pp. 1271–1275.

    MATH  Google Scholar 

  6. Mansimov, K.B., Integral necessary conditions for optimality of quasisingular controls in Goursat–Darboux systems, Autom. Remote Control, 1993, vol. 54, no. 5, pp. 732–739.

    MathSciNet  MATH  Google Scholar 

  7. Mansimov, K.B. and Mardanov, M.Dzh., Kachestvennaya teoriya optimal’nogo upravleniya sistemami Gursa–Darbu (Qualitative Theory of Optimal Control of Goursat–Darboux Systems), Baku: Elm, 2010.

    Google Scholar 

  8. Ermol’ev, Yu.M., Gulenko, V.P., and Tsarenko, T.I., Konechno-raznostnyi metod v zadachakh optimal’nogo upravleniya (Finite-Difference Method in Optimal Control Problems), Kiev: Naukova Dumka, 1978.

    MATH  Google Scholar 

  9. Rachinskii, V.V., Vvedenie v obshchuyu teoriyu dinamiki sorbtsii i khromatografii (Introduction to the General Theory of Sorption Dynamics and Chromatography), Moscow: Nauka, 1964.

    Google Scholar 

  10. Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1972.

    Google Scholar 

  11. Shaikhet, L.E., On optimal control of a class of stochastic partial differential equations, Mat. Zametki, 1982, no. 6, pp. 933–936.

  12. Shaikhet, L.E., Optimal control of certain hyperbolic and integral stochastic equations, J. Sov. Math., 1991, no. 4, pp. 457–462.

  13. Shaikhet, L.E., About an unsolved optimal control problem for stochastic partial differential equation, XVI Int. Conf. Dyn. Syst. Model. Stab. Invest. (Kiev, 2013), pp. 332–334.

  14. Qi Lu and Xu Zhang, Control theory for stochastic distributed parameter systems, an engineering perspective, Annu. Rev. Control, 2021, vol. 51, pp. 268–330.

    Article  MathSciNet  Google Scholar 

  15. Qi Lu and Xu Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Berlin: Springer, 2021.

    MATH  Google Scholar 

  16. Mastaliev, R.O., Necessary first-order optimality conditions in stochastic Goursat–Darboux systems, Dal’nevost. Mat. Zh., 2021, vol. 21, no. 1, pp. 89–104.

    MATH  Google Scholar 

  17. Walsh, J.B., An Introduction to Stochastic Partial Differential Equations, Berlin: Springer-Verlag, 1986.

    Book  Google Scholar 

  18. Ponomarenko, L.L., Stochastic infinite-dimensional Goursat problem, in Matematicheskii analiz i teoriya veroyatnostei (Mathematical Analysis and Probability Theory), Kiev, 1978, pp. 140–143.

  19. Mansimov, K.B. and Mastaliev, R.O., Representation of the solution of the Goursat problem for linear stochastic hyperbolic differential equations of the second order, Izv. Irkutsk. Gos. Univ. Ser. Mat., 2021, vol. 36, no. 2, pp. 29–43.

    MathSciNet  MATH  Google Scholar 

  20. Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V., Optimal’noe upravlenie (Optimal Control), Moscow: Nauka, 1979.

    MATH  Google Scholar 

  21. Gabasov, R. and Kirillova, F.M., Printsip maksimuma v teorii optimal’nogo upravleniya (The Maximum Principle in Optimal Control Theory), Moscow.: URSS, 2011.

    Google Scholar 

  22. Gabasov, R. and Kirillova, F.M., Osobye optimal’nye upravleniya (Singular Optimal Controls), Moscow: URSS, 2011.

    Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors express their deep gratitude to the referees for valuable comments on the content of the article.

Author information

Authors and Affiliations

  1. Baku State University, Baku, AZ1148, Azerbaijan

    K. B. Mansimov

  2. Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, AZ1141, Azerbaijan

    K. B. Mansimov & R. O. Mastaliev

Authors
  1. K. B. Mansimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. O. Mastaliev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to K. B. Mansimov or R. O. Mastaliev.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mansimov, K.B., Mastaliev, R.O. Necessary Optimality Conditions for Singular Controls in Stochastic Goursat–Darboux Systems. Autom Remote Control 83, 536–547 (2022). https://doi.org/10.1134/S0005117922040038

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117922040038

Keywords

Search

Navigation