Advertisement

Nonlinear Programming Methods for Solving Optimal Control Problems

  • Carla Durazzi
  • Emanuele Galligani
Chapter
  • 441 Downloads
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 58)

Abstract

This paper concerns with the problem of solving optimal control problems by means of nonlinear programming methods. The technique employed to obtain a mathematical programming problem from an optimal control problem is explained and the Newton interior-point method, chosen for its solution, is presented with special regard to the choice of the involved parameters. An analysis of the behaviour of the method is reported on four optimal control problems, related to improving water quality in an aeration process and to the study of diffusion convection processes.

Keywords

Optimal Control Nonlinear Programming Newton Interior-Point Method Finite Difference Approximations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Anderson, et al., LAPACK Users’ Guide. SIAM, Philadelphia, 1992.Google Scholar
  2. [2]
    S. Bellavia, Inexact interior-point method, Journal of Optimization Theory and Applications 96 (1998), 109–121.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    P. Benner, Computational methods for linear-quadratic optimization, Proceedings of Numerical Methods in Optimization, June 1997, Cortona, Italy. Rendiconti del Circolo Matematico di Palermo Ser. II, Suppl. 58 (1999), 21–56.MathSciNetzbMATHGoogle Scholar
  4. [4]
    P.T. Boggs, A.J. Kearsley, J.W. Tolle, A practical algorithm for general large scale nonlinear optimization problems, SIAM Journal on Optimization 9 (1999), 755–778.MathSciNetGoogle Scholar
  5. [5]
    M.D. Canon, C.D. Cullum, E. Polak, Theory of Optimal Control and Mathematical Programming, Mc Graw-Hill, New York, 1970.Google Scholar
  6. [6]
    C.H. Choi, A.J. Laub, Efficient matrix-valued algorithms for solving stiff Riccati differential equations, Proceedings of the 28th Conference on Decision and Control, December 1989, Tampa, Florida, 885–887.Google Scholar
  7. [7]
    R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods, SIAM Journal on Numerical Analysis 19 (1982), 400–408.CrossRefMathSciNetGoogle Scholar
  8. [8]
    J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, 1983.Google Scholar
  9. [9]
    C.R. Dohrmann, R.D. Robinett, Dynamic programming method for constrained discrete-time optimal control, Journal of Optimization Theory and Applications 101 (1999), 259–283.CrossRefMathSciNetGoogle Scholar
  10. [10]
    C. Durazzi, On the Newton interior-point method for nonlinear programming problems, Journal of Optimization Theory and Applications 104 (2000), 73–90.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    S.C. Eisenstat, H.F. Walker, Globally convergent inexact Newton methods, SIAM Journal on Optimization 4 (1994), 393–422.CrossRefMathSciNetGoogle Scholar
  12. [12]
    A.S. El-Bakry, R.A. Tapia, T. Tsuchiya, Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming, Journal of Optimization Theory and Applications 89 (1996), 507–541.CrossRefMathSciNetGoogle Scholar
  13. [13]
    T.M. El-Gindy, H.M. El-Hawary, M.S. Salim, M. El-Kady, A Chebyshev approximation for solving optimal control problems, Computers & Mathematics with Applications 29 (1995), 35–45.CrossRefMathSciNetGoogle Scholar
  14. [14]
    C.A. Floudas, Deterministic Global Optimization in Design, Control and Computational Chemistry, IMA Proceedings: Large Scale Optimization with Applications. Part II: Optimal Design and Control, Biegler LT, Conn A, Coleman L, Santosa F, (editors) 93 (1997), 129–187.Google Scholar
  15. [15]
    W. Hager, P.M. Pardalos (editors), Optimal Control: Theory, Algorithms and Applications, Kluwer Academic Publishers, 1998.Google Scholar
  16. [16]
    W. Hullet, Optimal estuary aeration: an application of distributed parameter control theory, Applied Mathematics and Optimization 1 (1974), 20–63.MathSciNetGoogle Scholar
  17. [17]
    C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995.Google Scholar
  18. [18]
    D.G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, Reading, 1984.Google Scholar
  19. [19]
    J.M. Ortega, Numerical Analysis: A Second Course, Academic Press, New York, 1972.Google Scholar
  20. [20]
    E.R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, Oxford, 1993.Google Scholar
  21. [21]
    H.W. Streeter, E.B. Phelps, A study of the pollution and natural purification of the Ohio river, U.S. Public Health Bulletin 1925; n. 146.Google Scholar
  22. [22]
    R.S. Varga, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, 1962.Google Scholar
  23. [23]
    J. Vlassenbroeck, R. Van Dooren, A Chebyshev technique for solving nonlinear optimal control problems. IEEE Transactions on Automatic Control 33 (1988), 333–340.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Carla Durazzi
    • 1
  • Emanuele Galligani
    • 2
  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItalia
  2. 2.Dipartimento di MatematicaUniversità di Modena e Reggio EmiliaModenaItalia

Personalised recommendations