Computational methods in Hilbert space for optimal control problems with delays

  • Andrzej P. Wierzbicki
  • Andrzej Hatko
Optimal Control
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3)


The paper consits of two parts. The first part is devoted to basic relations in the abstract theory of optimization and their relevance for computational methods. The concepts of the abstract theory (developed by Hurwicz, Uzawa, Dubovitski, Milyutin, Neustadt and others) linked together with the notion of a projection on a cone result in an unifying approach to computational methods of optimization. Several basic computational concepts, such as penalty functional techniques, problems of normality of optimal solutions, gradient projection and gradient reduction techniques, can be investigated in terms of a projection on a cone.

The second part of the paper presents an application of the gradient reduction technique in Hilbert space for optimal control problems with delays. Such an approach results in a family of computational methods, parallel to the methods known for finite-dimmensional and other problems: conjugate gradient methods, variable operator (variable metric) methods and generalized Newton's (second variation) method can be formulated and applied for optimal control problems with delays. The generalized Newton's method is, as usually, the most efficient; however, the computational difficulties in inverting the hessian operator are limiting strongly the applications of the method. Of other methods, the variable operator technique seems to be the most promissing.


Hilbert Space Lagrange Multiplier Optimal Control Problem Convex Cone Conjugate Gradient Method 
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. [1]
    Dubovitski A.J., Milyutin A.A.: Extremal problems with constraints. Journal of Computational Mathematics and Mathematical Physics (Russian), Vol V, No 3, p. 395–453, 1965.Google Scholar
  2. [2]
    Neustadt L.W.: An abstract variational theory with applications to a broad class of optimization problems. SIAM Journal on Control, Vol. V, No 1, p. 90–137, 1967.CrossRefGoogle Scholar
  3. [3]
    Goldshtein J.G.: Duality theory in mathematical programming (Russian), Nauka, Moscow 1971.Google Scholar
  4. [4]
    Pshenitshny B.N.: Necessary conditions of optimality (Russian), Nauka, Moscow 1969.Google Scholar
  5. [5]
    Luenberger D.G.: Optimization by vector space methods. J. Wiley, N. York 1969.Google Scholar
  6. [6]
    Neustadt L.W. A general theory of extremals. Journal on Computer and System Science, Vol. III, p. 57–91, 1969.Google Scholar
  7. [7]
    Girsanov I.W.: Lectures on mathematical theory of extremal problems. University of Moscow, 1970.Google Scholar
  8. [8]
    Wierzbicki A.P.: Maximum principle for semiconvex performance functionals. SIAM Journal on Control, V. X No 3 p. 444–459, 1972.CrossRefGoogle Scholar
  9. [9]
    Galperin A.M.: Towards the theory of permissible directions (Russian), Kibernetika No 2, p. 51–59, 1972.Google Scholar
  10. [10]
    Fiacco, A.V., Mc Cormick G.P.: The sequential unconstrained minimization technique for nonlinear programming. Management Science. Vo. X, No 2, p. 360–366, 1964.Google Scholar
  11. [11]
    Balakrishnan, A.V. A computational approach to the maximum principle. Journal of Computer and System Science, Vol. V, 1971.Google Scholar
  12. [12]
    Wierzbicki A.P.: A penalty function shifting method in constrained static optimization and its convergence properties. Archiwum Automatyki i Telemechaniki, Vol. XVI, No 4, p. 395–416, 1971.Google Scholar
  13. [13]
    Powell, M.J.D.: A method for nonlinear constraints in minimisation problems. In R. Fletcher: Optimization, Academic Press, N. York 1969.Google Scholar
  14. [14]
    Rosen, J.B.: The gradient projection method for nonlinear programming. Part I, II. Journal of SIAM, Vol. VIII, p. 181–217, 1960, Vol. IX, p. 514–532, 1962.Google Scholar
  15. [15]
    Wolfe, P. Methods of nonlinear programming. In J. Abadie: Nonlinear Programming, Interscience, J. Wiley, N. York, 1967.Google Scholar
  16. [16]
    Horwitz L.B., Sarachik P.E.: Davidon's method in Hilbert Space. SIAM J. on Appl. Math., Vol. XVI, No 4, p. 676–695, 1968.CrossRefGoogle Scholar
  17. [17]
    Wierzbicki A.P.: Coordination and sensitivity analysis of a large scale problem with performance iteration. Proc. of V-th Congress of IFAC, Paris 1972.Google Scholar
  18. [18]
    Wierzbicki A.P.: Methods of mathematical programming in Hilbert space. Polish-Italian Meeting on Control Theory and Applications, Cracow 1972.Google Scholar
  19. [19]
    Chan H.C., Perkins W.R. Optimization of time delay systems using parameter imbedding. Proc. of V-th Congress of IFAC, Paris 1972.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Andrzej P. Wierzbicki
  • Andrzej Hatko
    • 1
  1. 1.Institute of Automatic Control, Faculty of ElectronicsTechnical University of WarsawWarsawPoland

Personalised recommendations