Numerical Optimal Control Via Smooth Penalty Functions

  • Mohammed Hasan
  • Bruce N. Lundberg
  • Aubrey B. Poore
  • Bing Yang
Part of the Progress in Systems and Control Theory book series (PSCT, volume 1)


The use of first order necessary conditions to solve optimal control problems has two principal drawbacks: a sufficiently close initial approximation is required to ensure local convergence and this initial approximation must be cHosen so that the convergence is to a local optimum. We present a class of algorithms which resolves both of these difficulties and which is ultimately based on the solution of first order necessary conditions. The key ingredients are three smooth penalty functions (the quadratic penalty for equality constraints and the log barrier or quadratic loss for inequality constraints), a parameterized system of nonlinear equations, and efficient predictor-corrector continuation techniques to follow the penalty path to optimality. This parameterized system of equations is essentially a homotopy which is derived from these penalty functions, contains the penalty path as a solution, and represents a perturbation of the first order necessary conditions. However, it differs significantly from Homotopies for nonlinear equations in that an unconstrained optimization technique is required to obtain an initial point.


Optimal Control Problem Penalty Function Sequential Quadratic Programming Continuation Method Pontryagin Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [01]
    A.V. Balakrishnan and L. W. Neustadt, eds., Computing MetHods in Optimization Problems, Academic Press, New York, 1964.zbMATHGoogle Scholar
  2. [02]
    R. E. Bellman and S. E. Dreyfus, Applied Dynamic Programming, Princeton University Press, Princeton, 1962.zbMATHGoogle Scholar
  3. [03]
    A. B. Bryson Jr. and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere Publishing Corporation, Washington, D.C., 1975.Google Scholar
  4. [04]
    L. Cesari, Optimization Theory and Applications, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar
  5. [05]
    A. V. Fiacco and G. P. McCormik, Nonlinear Sequential Unconstrained Minimization Techniques, John Wiley and Sons, Inc., New York, 1968.zbMATHGoogle Scholar
  6. [06]
    R. Fletcher, Practical MetHods of Optimization, Second Edition, John Wiley k Sons Ltd., New York, 1987.zbMATHGoogle Scholar
  7. [07]
    I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.Google Scholar
  8. [08]
    S. T. Glad, “A combination of penalty function and multiplier metHods for solving optimal control problems”, J. Opt. Th. Applic, v. 28 (1979), 303–329.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [09]
    A. GRIEWANK and Ph. L. TOINT, “Numerical experiments with partially separable optimization problems”, in D. F. Griffiths, ed., Nu¬merical Analysis, Dundee 1983, Lecture Notes in Mathematics 1066, Springer-Verlag, Berlin, 1984.Google Scholar
  10. [10]
    W. A. Gruver and E. Sachs, Algorithmic Methods in Optimal Control, Pitman Publishing Inc., London, 1980.Google Scholar
  11. [11]
    M. HASAN and A. B. Poore, “A bifurcation analysis of the quadratic penalty-log barrier function”, in preparation.Google Scholar
  12. [12]
    L. Hasdorff, Gradient Optimization and Nonlinear Control, John Wiley k Sons, New York, 1976.Google Scholar
  13. [13]
    H.B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems”, Applications of Bifurcation Theory (P. Rabinowitz, ed.), Academic Press, New York, 1977, pp. 359–384.Google Scholar
  14. [14]
    C. T. KELLEY and E. W. Sachs, “Quasi-Newton metHods and unconstrained optimal control problems”, SIAM J. Control and Optimization, v. 25 (1987), 1503–1515.Google Scholar
  15. [15]
    D. KRAFT, “On converting optimal control problems into nonlinear programming problems”, Computational Mathematical Programming, NATO ASI series, vol F15 (K. Schittkowski, ed.), Springer-Verlag, Berlin, 1985.Google Scholar
  16. [16]
    E. B. LEE and L. MARKUS, Foundations of Optimal Control Theory, John Wiley and Sons, Inc., New York, 1967.Google Scholar
  17. [17]
    B. N. LUNDBERG, A. B. Poore and B. YANG, “Smooth penalty functions and continuation metHods for constrained optimization”, to appear in Lectures in Applied Mathematics.Google Scholar
  18. [18]
    B. N. LUNDBERG and A. B. Poore, “Variable order Adams-Bash-forth predictors with error-stepsize control in continuation metHods”, submitted for publication, 1988.Google Scholar
  19. [19]
    I. H. Mufti, Computational Methods in Optimal Control Problems, Springer-Ver lag, New York 1970.zbMATHGoogle Scholar
  20. [20]
    L. W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, New Jersey, 1976.zbMATHGoogle Scholar
  21. [21]
    A. B. Poore and Q. Al-Hassan, “The expanded Lagrangian system for constrained optimization problems”, SIAM J. Control and Optimization, v. 26 (1988), 417–427.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    A. B. Poore and C. A. Tiahrt, “Bifurcation problems in nonlinear parametric programming”, Mathematical Programming, v. 39 (1987), 189–205.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    K. L. Teo and L. T. Yeo, “On the computational metHods of optimal control problems”, Int. J. Systems Sei., v. 10 (1979), 51–76.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    V. M. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems, John Wiley &, Sons, New York, 1986.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Mohammed Hasan
    • 1
  • Bruce N. Lundberg
    • 1
  • Aubrey B. Poore
    • 1
  • Bing Yang
    • 1
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA

Personalised recommendations