Skip to main content
Springer Nature Link
Log in

The Critical Cone and Second-Order Optimality Conditions for a State-Constrained Optimal Control Problem

  • Conference paper
  • First Online:
Numerical Computations: Theory and Algorithms (NUMTA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14476))

  • 158 Accesses

Abstract

A state-constrained optimal control problem is considered in which the starting position is fixed and the terminal position is free. The endpoints are strictly included within the set shaped by the given state constraint which is considered scalar. For this problem formulation, the second-order necessary optimality conditions are derived including the Legendre condition over the critical cone.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Rampazzo, F., Vinter, R.B.: A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control. IMA J. Math. Control. Inf. 16(4), 335–351 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Khalil, N.: Optimality conditions for optimal control problems and applications. Optimization and Control. PhD thesis, Universite de Bretagne occidentale (2017)

    Google Scholar 

  3. Karamzin, D.Yu. and Pereira, F.L.: Some Remarks on the Issue of Normality in State-constrained Optimal Control Problems.In: 2022 IEEE 61st Conference on Decision and Control (CDC), Cancun, Mexico, pp. 552–557 (2022)

    Google Scholar 

  4. Karamzin, D.Y.: Normality and Second-order Optimality Conditions in State-constrained Optimal Control Problems with Bounded Minimizers. J. Differ. Equ. 366, 378–407 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arutyunov, A.V., Aseev, S.M.: State constraints in Optimal Control – the degeneracy phenomenon, System Control Letters, 26 (1995)

    Google Scholar 

  6. Ferreira, M.M. and Vinter, R.B.: When is the maximum principle for state constrained problems nondegenerate? J. Math. Anal. and Appl. 187 (1994)

    Google Scholar 

  7. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Investigation of controllability and regularity conditions for state constrained problems. IFAC-PapersOnLine 50(1), 6295–6302 (2017)

    Article  MATH  Google Scholar 

  8. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: Maximum principle and second-order optimality conditions in control problems with mixed constraints. Axioms 11(2), 40 (2022)

    Article  MATH  Google Scholar 

  9. Arutyunov, A., Karamzin, D.: A survey on regularity conditions for state-constrained optimal control problems and the non-degenerate maximum principle. J. Optim. Theory Appl. 184(3), 697–723 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arutyunov, A.V., Karamzin, D.Y.: On some continuity properties of the measure lagrange multiplier from the maximum principle for state constrained problems. SIAM J. Control. Optim. 53, 4 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Arutyunov, A.V.: Perturbations of extremal problems with constraints and necessary optimality conditions. J. Sov. Math. 54, 6 (1991)

    Article  MATH  Google Scholar 

  13. Russak, I.B.: Second order necessary conditions for problems with state inequality constraints. SIAM J. Control 13(2), 372–388 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the Russian Science Foundation (project N 20-11-20131, https://rscf.ru/en/project/20-11-20131/) and carried out at V.A. Trapez-nikov Institute of Control Sciences of RAS.

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, Russia

    Dmitry Karamzin

Authors
  1. Dmitry Karamzin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Karamzin .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

  3. University of Calabria, Rende, Italy

    Annabella Astorino

Rights and permissions

Reprints and permissions

Copyright information

© 2025 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Karamzin, D. (2025). The Critical Cone and Second-Order Optimality Conditions for a State-Constrained Optimal Control Problem. In: Sergeyev, Y.D., Kvasov, D.E., Astorino, A. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2023. Lecture Notes in Computer Science, vol 14476. Springer, Cham. https://doi.org/10.1007/978-3-031-81241-5_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-81241-5_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-81240-8

  • Online ISBN: 978-3-031-81241-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics