Skip to main content
SpringerLink
Log in

Necessary first and second order optimality conditions in a discrete optimal control problem

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

An optimal control problem described by a system of two-dimensional nonlinear difference Volterra-type equations that are a discrete analogue of the two-dimensional integral Volterra equation is considered. The first and second variations of the quality functional are calculated assuming that the control domain is open. They are used to obtain an analogue of the Euler equation and to find the necessary second order optimality conditions that can be checked constructively.

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. Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi para-metrami (Theory of Optimal Control of Systems with Distributed Parameters), Moscow: Nauka, 1965.

    Google Scholar 

  2. Sheng-Chao, H., Optimal Control Problems with a System of Integral Equations and Restricted Phase Coordinates, Siam J. Control, 1972, vol. 10, pp. 14–36.

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlson, D.A., An Elementary Proof of Maximum Principle for Optimal Control Problems Governed by Volterra Integral Equations, J. Opt. Theory Appl., 1987, vol. 54, pp. 113–124.

    Article  Google Scholar 

  4. Chikrii, A.A., Chikrii, G.Ts., and Volyanskii, K.Yu., Quasilinear Positional Integral Games of Approach, J. Autom. Inform. Sci., 2001, vol. 33, no. 10, p. 22.

    Google Scholar 

  5. Glushkov, V.M., Ivanov, V.V., and Yanenko, V.M., Modelirovanie razvivayushchikhsya sistem (Modeling of Developing Systems), Moscow: Nauka, 1983.

    MATH  Google Scholar 

  6. Abdullaev, A.A. and Mansimov, K.B., Second-Order Necessary Conditions of Optimality for Processes Described by a System of Two-Dimensional Nonlinear Integral Equations of the Volterra Type, Automat. Remote Cont., 2000, vol. 61, pp. 1–8.

    MathSciNet  MATH  Google Scholar 

  7. Kolmanovskii, V.B., On Asymptotic Equivalence of the Solutions of Some Volterra Difference Equations, Automat. Remote Cont., 2001, vol. 62, pp. 548–556.

    Article  MathSciNet  MATH  Google Scholar 

  8. Ivinskaya, E.V. and Kolmanovskii, V.B., On Limitedness of the Solutions of Some Difference Volterra Equations, Automat. Remote Cont., 2000, vol. 61, pp. 1317–1327.

    MathSciNet  MATH  Google Scholar 

  9. Abuladze, A.A., Zadachi optimal’nogo upravleniya dlya sistem opisyvaemykh integral’nymi uravneniyami (Optimal Control Problems for Systems Described by Integral Equations), Tbilisi: Tbil. Gos. Univ., 1988.

    MATH  Google Scholar 

  10. Dymkov, M.P., Ekstremal’nye zadachi v mnogoparametricheskikh sistemakh upravleniya (Extreme Problems in Multiparametrical Control Systems), Minsk: Belor. Gos. Ekonom. Univ., 2005.

    Google Scholar 

  11. Dymkov, M.P., Optimal Control of Volterra Discrete System According to Quadratic Functional, Dokl. Akad. Nauk Belorusi, 1997, vol. 41, no. 3, pp. 10–16.

    MathSciNet  MATH  Google Scholar 

  12. Kolmanovskii, V.B., On Asymptotic Properties of Solutions of Some Volterra Nonlinear Systems, Automat. Remote Cont., 2000, vol. 61, pp. 577–584.

    MathSciNet  Google Scholar 

  13. Choi, S.K., Goo, Y.H., and Koo, N.J., Asymptotic Behavior of Nonlinear Volterra Difference Systems, Bull. Korean Math. Soc., 2007, vol. 44, no. 1, pp. 177–184.

    Article  MathSciNet  MATH  Google Scholar 

  14. Zouyonsefain, M. and Leela, S., Stability Results for Difference Equations of Volterra Type, Appl. Math. Comput., 1990, vol. 36, no. 1, pp. 51–61.

    Article  MathSciNet  Google Scholar 

  15. Choi, S.K., Goo, Y.H., and Koo, N.J., Existence and Boundedness of Solution for Volterra Discrete Equations, J. Chungcheong Math. Soc., 2006, vol. 19, pp. 237–244.

    Google Scholar 

  16. Mansimov, K.B., Diskretnye sistemy (Discrete Systems), Baku: Bakin. Gos. Univ., 2002.

    Google Scholar 

  17. Gabasov, R. and Kirillova, F.M., Osobye optimal’nye upravleniya (Special Optimal Controls), Moscow: Nauka, 1973.

    Google Scholar 

  18. Dem’yanov, V.F., Usloviya ekstremuma i variatsionnoe ischislenie (Conditions of Extremum and Variation Calculation), Moscow: Vysshaya Shkola, 2005.

    Google Scholar 

  19. Choi, S.K. and Koo, N.J., Asymptotic Property of Linear Volterra Difference Systems, J. Math. Anal. Appl., 2006, vol. 321, no. 1, pp. 260–272.

    Article  MathSciNet  MATH  Google Scholar 

  20. Choi, S.K., Koo, N.J., and Goo, Y.H., Asymptotic Property of Nonlinear Volterra Difference Systems, Nonlinear Anal. Ser. A: Theory Methods, 2002, vol. 51, pp. 321–337.

    Article  MathSciNet  MATH  Google Scholar 

  21. Mansimov, K.B., Optimization of One Class of Duscrete Two-Parametric Systems, Differ. Uravn. Ikh Primen., 1991, pp. 358–361.

    Google Scholar 

  22. Mansimov, K.B., Necessary Conditions of Spacial Process Optimality in Optimal Control Problems, Doctoral (Phys.-Mat.) Dissertation, Baku: Baku State Univ., 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Cybernetics, National Academy of Sciences of Azerbaijan, Baku State University, ul. B. Vagabzade 9, Baku, Az-1141, Azerbaijan

    R. R. Amirova & K. B. Mansimov

Authors
  1. R. R. Amirova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. B. Mansimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. B. Mansimov.

Additional information

Original Russian Text © R.R. Amirova, K.B. Mansimov, 2013, published in Avtomatika i Vychislitel’naya Tekhnika, 2013, No. 1, pp. 13–22.

About this article

Cite this article

Amirova, R.R., Mansimov, K.B. Necessary first and second order optimality conditions in a discrete optimal control problem. Aut. Control Comp. Sci. 47, 7–14 (2013). https://doi.org/10.3103/S0146411613010021

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411613010021

Keywords

Search

Navigation