Skip to main content
SpringerLink
Log in

On Estimates of the Solutions of Inverse Problems of Optimal Control

  • Conference paper
  • First Online:
Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1090))

Abstract

This paper is devoted to the problem of reconstruction of the normal control generating a realized trajectory of a dynamic control system by using known inaccurate measurements of this trajectory. A class of dynamic control systems with dynamics linear in controls and non-linear in state coordinates is considered. A new method, suggested in earlier publications, for solving such problems is discussed. This approach relies on necessary optimality conditions in an auxiliary variational problem on extremum of an integral discrepancy functional. The distinguishing feature of the method is using a functional which is convex in control variables and concave in state variables discrepancy. This form of the functional allows to obtain oscillating solutions. In this paper the estimates of the error of the discussed method are exposed and validated.

This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00074) and by the Ural Branch of the Russian Academy of Sciences (project no. 18-1-1-10).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Blizorukova, M.: On the reconstruction of the trajectory and control in a nonlinear second-order system. Proc. Steklov Inst. Math. 275(1), 1–11 (2011). https://doi.org/10.1134/S008154381109001X

    Article  MathSciNet  MATH  Google Scholar 

  2. Krasovskii, N.: Teoriya upravleniya dvizheniem [Theory of motion control]. Nauka, Moscow (1968). (in Russian)

    Google Scholar 

  3. Krasovskii, N., Subbotin, A.: Positcionnie differentcialnie igri [Positional differential games]. Nauka, Moscow (1974). (in Russian)

    Google Scholar 

  4. Krupennikov, E.: A new approximate method for construction of the normal control. IFAC-PapersOnLine 51(32), 343–348 (2018). https://doi.org/10.1016/j.ifacol.2018.11.407

    Article  Google Scholar 

  5. Krupennikov, E.: On control reconstruction problems for dynamic systems linear in controls. In: Static & Dynamic Game Theory Foundations & Applications. Birkhauser, Cham (2018)

    Google Scholar 

  6. Krupennikov, E.: Solution of inverse problems for control systems with large control parameter dimension. IFAC-PapersOnLine 51(32), 434–438 (2018). https://doi.org/10.1016/j.ifacol.2018.11.423

    Article  Google Scholar 

  7. Kryazhimskij, A., Osipov, Y.: Modelling of a control in a dynamic system. Eng. Cybern. 21(2), 38–47 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Magnus, J., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd edn. Wiley, Hoboken (2019). https://doi.org/10.1002/9781119541219

    Book  MATH  Google Scholar 

  9. Osipov, Y., Kryazhimskii, A.: Inverse Problems for Ordinary Differential Equations: Dynamical Solutions. Gordon and Breach, London (1995)

    MATH  Google Scholar 

  10. Osipov, Y., Kryazhimskii, A., Maksimov, V.: Some algorithms for the dynamic reconstruction of inputs. Proc. Steklov Inst. Math. 275(1), 86–120 (2011). https://doi.org/10.1134/S0081543811090082

    Article  MathSciNet  MATH  Google Scholar 

  11. Subbotina, N., Krupennikov, E.: Dynamic programming to identification problems. World J. Eng. Technol. 4(3B), 228–234 (2016)

    Article  Google Scholar 

  12. Subbotina, N., Tokmantsev, T., Krupennikov, E.: On the solution of inverse problems of dynamics of linearly controlled systems by the negative discrepancy method. Proc. Steklov Inst. Math. 291, 253–262 (2015). https://doi.org/10.1134/S0081543815080209

    Article  MathSciNet  MATH  Google Scholar 

  13. Subbotina, N., Tokmantsev, T., Krupennikov, E.: Dynamic programming to reconstruction problems for a macroeconomic model. In: IFIP Advances in Information and Communication Technology. [S.l.], vol. 494, pp. 472–481 (2017)

    Google Scholar 

  14. Tikhonov, A.: Ob ustoichivosti obratnih zadach [on the stability of inverse problems]. Doklady Academii Nauk SSSR 39, 195–198 (1943). (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences (IMM UB RAS), 16 S. Kovalevskaya Str., Ekaterinburg, 620990, Russia

    Evgenii A. Krupennikov

  2. Ural Federal University named after the first President of Russia B.N.Yeltsin, 19 Mira Street, Ekaterinburg, 620002, Russia

    Evgenii A. Krupennikov

Authors
  1. Evgenii A. Krupennikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgenii A. Krupennikov .

Editor information

Editors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Igor Bykadorov

  2. University of Greenwich, London, UK

    Vitaly Strusevich

  3. University of Aveiro, Aveiro, Portugal

    Tatiana Tchemisova

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Krupennikov, E.A. (2019). On Estimates of the Solutions of Inverse Problems of Optimal Control. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_39

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-33394-2_39

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-33393-5

  • Online ISBN: 978-3-030-33394-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation