Skip to main content
SpringerLink
Log in

Stability of solutions to control constrained nonlinear optimal control problems

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

We consider a family of nonlinear optimal control problems depending on a parameter. Under the assumption of a second-order sufficient optimality condition it is shown that the solutions of the problems as well as the associated Lagrange multipliers are Lipschitz continuous functions of the parameter.

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. Alt W., On the approximation of infinite optimization problems with an application to optimal control problems, Appl. Math. Optim. 12 (1984), 15–27.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubin J. P., Cellina A., Differential Inclusions, Springer-Verlag, Berlin, 1984.

    Book  MATH  Google Scholar 

  3. Bauer H., Wahrscheinlichkeitstheorie und Grudzüge der Maßtheorie, Gruyter, Berlin, 1974.

    Book  MATH  Google Scholar 

  4. Dontchev A. L., Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems, Lecture Notes in Control and Information Sciences, Vol. 52, Springer-Verlag, Berlin, 1983.

    Book  Google Scholar 

  5. Hager W. W., Lipschitz continuity for constrained processes, SIAM J. Control Optim. 17 (1979), 321–337.

    Article  MathSciNet  MATH  Google Scholar 

  6. Hermes H., Lasalle J. P., Functional Analysis and Time Optimal Control, Academic Press, New York, 1969.

    MATH  Google Scholar 

  7. Hestenes M. R., Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966.

    MATH  Google Scholar 

  8. Ioffe A. D., Necessary and sufficient conditions for a local minimum. 3: Seocnd-order conditions and augmented duality, SIAM J. Control Optim. 17 (1979), 266–288.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ioffe A. D., Tihomirov V. M., Theory of Extremal Problems, North Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  10. Kirsch A., Warth W., Werner J., Notwendige Optimalitätsbedingungen und ihre Anwendung, Springer-Verlag, Berlin, 1978.

    Book  MATH  Google Scholar 

  11. Malanowski K., On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints, Appl. Math. Optim. 12 (1984), 231–245.

    Article  MathSciNet  MATH  Google Scholar 

  12. Malanowski K., Stability and sensitivity of solutions to optimal control problems for systems with control appearing linearly, Appl. Math. Optim. 16 (1987), 73–91.

    Article  MathSciNet  MATH  Google Scholar 

  13. Maurer H., First-and second-order sufficient optimality conditions in mathematical programming and optimal control, Math. Programming Stud. 14 (1981), 163–177.

    Article  MathSciNet  MATH  Google Scholar 

  14. Robinson S. M., Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497–513.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bayreuth, Postfach 101251, D-8580, Bayreuth, West Germany

    Walter Alt

Authors
  1. Walter Alt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. Stoer

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alt, W. Stability of solutions to control constrained nonlinear optimal control problems. Appl Math Optim 21, 53–68 (1990). https://doi.org/10.1007/BF01445157

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01445157

Keywords

Search

Navigation