Skip to main content
SpringerLink
Log in

Solution differentiability for parametric nonlinear control problems with control-state constraints

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC.

Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.

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.,Parametric Optimization with Applications to Optimal Control and Sequential Quadratic Programming, Report No. 35, Bayreuther Mathematische Schriften, University of Bayreuth, Bayreuth, Germany, 1991.

    Google Scholar 

  2. Colonius, F., andKunisch, K.,Sensitivity Analysis for Optimization Problems in Hilbert Spaces with Bilateral Constraints, Report No. 267, Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung der Deutschen Forschungsgemeinschaft, University of Augsburg, Augsburg, Germany, 1991.

    Google Scholar 

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

    Google Scholar 

  4. Dontchev, A. L., Hager, W. W., Poore, A. B., andYang, B.,Optimality, Stability, and Convergence in Nonlinear Control, Applied Mathematics and Optimization, Vol. 31, pp. 297–326, 1995.

    Article  Google Scholar 

  5. Ito, K., andKunisch, K.,Sensitivity Analysis of Solutions to Optimization Problems in Hilbert Spaces with Applications to Optimal Control and Estimation, Journal of Differential Equations, Vol. 99, pp. 1–40, 1992.

    Article  Google Scholar 

  6. Malanowski, K.,Sensitivity Analysis of Optimization Problems in Hilbert Space with Application to Optimal Control, Applied Mathematics and Optimization, Vol. 21, pp. 1–20, 1991.

    Google Scholar 

  7. Malanowski, K.,Second-Order Conditions and Constraint Qualifications in Stability and Sensitivity Analysis of Solutions to Optimization Problems in Hilbert Spaces, Applied Mathematics and Optimization, Vol. 25, pp. 51–79, 1992.

    Article  Google Scholar 

  8. Malanowski, K.,Stability and Sensitivity of Solutions to Nonlinear Optimal Control Problems, Preprint, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland, 1991.

    Google Scholar 

  9. Malanowski, K.,Regularity of Solutions in Stability Analysis of Optimization and Optimal Control Problems, Working Paper No. 1/93, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland, 1993.

    Google Scholar 

  10. Malanowski, K.,Two-Norm Approach in Stability and Sensitivity Analysis of Optimization and Optimal Control Problems, Advances in Mathematical Sciences and Applications, Vol. 2, pp. 397–443, 1993.

    Google Scholar 

  11. Maurer, H., andPesch, H. J.,Solution Differentiability for Parametric Nonlinear Control Problems, SIAM Journal on Control and Optimization, Vol. 32, pp. 1542–1554, 1994.

    Article  Google Scholar 

  12. Bock, H. G.,Zur numerischen Behandlung zustandsbeschränkter Steuerungsprobleme mit Mehrzielmethode und Homotopieverfahren, ZAMM, Vol. 57, pp. T266-T268, 1977.

    Google Scholar 

  13. Bock, H. G., andKrämer-Eis, P.,An Efficient Algorithm for Approximate Computation of Feedback Control Laws in Nonlinear Processes, ZAMM, Vol. 61, pp. T330-T332, 1981.

    Google Scholar 

  14. Krämer-Eis, P.,Ein Mehrzielverfahren zur numerischen Berechnung optimaler Feedback-Steuerungen bei beschränkten nichtlinearen Steuerungsproblemen, Bonner Mathematische Schriften No. 164, University of Bonn, Bonn, Germany, 1985.

    Google Scholar 

  15. Kugelmann, B., andPesch, H. J.,A New General Guidance Method in Constrained Optimal Control, Part 1: Numerical Method, Journal of Optimization Theory and Applications, Vol. 67, pp. 421–435, 1990.

    Article  Google Scholar 

  16. Kugelmann, B., andPesch, H. J.,A New General Guidance Method in Constrained Optimal Control, Part 2: Application to Space Shuttle Guidance, Journal of Optimization Theory and Applications, Vol. 67, pp. 437–446, 1990.

    Article  Google Scholar 

  17. Pesch, H. J.,Echtzeitberechnung fastoptimaler Rückkopplungssteuerungen mit Beschränkungen, Habilitationsschrift, Munich University of Technology, Munich, Germany, 1986.

    Google Scholar 

  18. Pesch, H. J.,Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part 1: Neighboring Extremals, Optimal Control Applications and Methods, Vol. 10, pp. 129–145, 1989.

    Google Scholar 

  19. Pesch, H. J.,Real-Time Computation of Feedback Controls for Constrained Optimal Control Problems, Part 2: A Correction Method Based on Multiple Shooting, Optimal Control Applications and Methods, Vol. 10, pp. 147–171, 1989.

    Google Scholar 

  20. Pesch, H. J.,Optimal Control Problems under Disturbances, Proceedings of the 14th IFIP Conference on System Modelling and Optimization, Leipzig, Germany, 1989; Edited by H. J. Sebastian and K. Tammer, Lecture Notes in Control and Information Science, Springer, Berlin, Germany, Vol. 143, pp. 377–386, 1990.

    Google Scholar 

  21. Orrell, D., andZeidan, V.,Another Jacobi Sufficiency Criterion for Optimal Control with Smooth constraints, Journal of Optimization Theory and Applications, Vol. 58, pp. 283–300, 1988.

    Article  Google Scholar 

  22. Zeidan, V.,Sufficiency Conditions with Minimal Regularity Assumption, Applied Mathematics and Optimization, Vol. 20, pp. 19–31, 1989.

    Article  Google Scholar 

  23. Pickenhain, S.,Sufficiency Conditions for Weak Local Minima in Multidimensional Optimal Control Problems with Mixed Control-State Restrictions, Zeitschrift für Analysis und Ihre Anwendungen, Vol. 11, pp. 559–568, 1992.

    Google Scholar 

  24. Maurer, H.,The Two-Norm Approach for Second-Order Sufficient Conditions in Mathematical Programming and Optimal Control, Report No. 6/92-N, Angewandte Mathematik und Informatik, University of Münster, Münster, Germany, 1992.

    Google Scholar 

  25. Maurer, H., andPickenhain, S.,Second-Order Sufficient Conditions for Optimal Control Problems with Mixed Control-State Constraints, Journal of Optimization Theory and Applications, Vol. 86, pp. 649–667, 1995.

    Article  Google Scholar 

  26. Miele, A., Pritchard, R. E., andDamoulakis, J. N.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 5, pp. 235–282, 1970.

    Article  Google Scholar 

  27. Miele, A., Iyer, R. R., andWell, K. H.,Modified Quasilinearization and Optimal Initial Choice of the Multipliers, Part 2: Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 6, pp. 381–409, 1970.

    Article  Google Scholar 

  28. Neustadt, L. W.,Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, New Jersey, 1976.

    Google Scholar 

  29. Zeidan, V.,Sufficiency Criteria via Focal Points and via Coupled Points, SIAM Journal Control and Optimization, Vol. 30, pp. 82–98, 1992.

    Article  Google Scholar 

  30. Maurer, H.,First- and Second-Order Sufficient Optimality Conditions in Mathematical Programming and Optimal Control, Mathematical Programming Study, Vol. 14, pp. 163–177, 1981.

    Google Scholar 

  31. Sorger, G.,Sufficient Optimality Conditions for Nonconvex Problems with State Constraints, Journal of Optimization Theory and Applications, Vol. 62, pp. 289–310, 1989.

    Article  Google Scholar 

  32. Reid, W. T.,Riccati Differential Equations, Academic Press, New York, New York, 1972.

    Google Scholar 

  33. Bulirsch, R.,Die Mehrzielmethode zur Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung, Report of the Carl-Cranz Gesellschaft, DLR, Oberpfaffenhofen, Germany, 1971.

    Google Scholar 

  34. Oberle, H. J., andGrimm, W.,BNDSCO—A Program for the Numerical Solution of Optimal Control Problems, Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.

    Google Scholar 

  35. Fan, L. T.,The Continuous Maximum Principle, John Wiley and Sons, New York, New York, 1966.

    Google Scholar 

Download references

Author information

Author notes

  1. H. J. Pesch (Professor of Mathematics)

    Present address: Institut für Mathematik, Technische Universität Clausthal, Clausthal-Zenerfeld, Germany

Authors and Affiliations

  1. Institut für Numerische und Instrumentelle Mathematik, Westfälische Wilhelms-Universität Münster, Münster, Germany

    H. Maurer (Professor of Mathematics)

  2. Mathematisches Institut, Technische Universität München, München, Germany

    H. J. Pesch (Professor of Mathematics)

Authors
  1. H. Maurer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. J. Pesch
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by R. Bulirsch

This paper is dedicated to Professor J. Stoer on the occasion of his 60th birthday.

The authors are indebted to K. Malanowski for helpful discussions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maurer, H., Pesch, H.J. Solution differentiability for parametric nonlinear control problems with control-state constraints. J Optim Theory Appl 86, 285–309 (1995). https://doi.org/10.1007/BF02192081

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02192081

Key Words

Search

Navigation