Volterra Series and Optimal Control

  • Michel Fliess
  • Françoise Lamnabhi-Lagarrigue
Part of the Mathematics and Its Applications book series (MAIA, volume 29)


After a review of some of the theory of non-commutative generating powers series expansions and their relationship with Taylor expansions of the Volterra kernels, we show that those kernels can be most naturally expressed by using the Hamiltonian of the system. This is applied to a general optimal control problem in order to get higher order necessary conditions for optimality. These systematically receive a Hamiltonian interpretation.


Functional Derivative Volterra Series Volterra Kernel Functional Expansion Generate Power Series 
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]
    A.A. AGRACEV and R.V. GAMKRELIDZE, Exponential representation of flows and the chronological calculus (en russe), Mat. Sbornik, 107, 1978, 467–532. Traduction anglaise: Math. USSR Sbornik, 35, 1978, 727–785.MathSciNetGoogle Scholar
  2. [2]
    D.J. BELL and D.H. JACOBSON, Singular Optimal Control Problems, Academic Press, London, 1975.zbMATHGoogle Scholar
  3. [3]
    S. BERGER, Nonlinearity and Functional Analysis. Academic Press, New York, 1977.zbMATHGoogle Scholar
  4. [4]
    N. BOURBAKI, Groupes et algèbres de Lie, Chap. 2 et 3, Hermann, Paris, 1972.Google Scholar
  5. [5]
    R.W. BROCKETT, Lie theory, functional expansions and necessary conditions in optimal control, in “Mathematical Control Theory” (W.A. Coppel ed.), Lect. Notes Math. 680, p. 68–76, Springer, Berlin, 1978.CrossRefGoogle Scholar
  6. [6]
    M.D. DONSKER and S.L. LIONS, Fréchet-Volterra variational equations, boundary value problems, and function space integrals, Acta Math., 108, 1962, 147–228.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    B. DUBROVIN, S. NOVIKOV and A. FOMENKO, Modern geometry, Springer, New York, 1984: French version: MIR, Moscou, 1982Google Scholar
  8. [8]
    M. FLIESS, Fonctionnelles causales non linéaires et indéterminées non commutatives, Bull. Soc. Math. France, 109, 1981, 3–40.MathSciNetzbMATHGoogle Scholar
  9. [9]
    M. FLIESS, Vers une notion de dérivation fonctionnelle causale, Ann. Inst. H. Poincaré. Anal. Non linéaire, to appear.Google Scholar
  10. [10]
    M. FLIESS, M. LAMNABHI and F. LAMNABHI-LAGARRIGUE, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits Systems, 30, 1983, 554–570.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    M. FLIESS and F. LAMNABHI-LAGARRIGUE, Séries de Volterra et formalisme hamiltonien, CR. Acad. Sc. Paris, I-299, 1984, 783–785.MathSciNetGoogle Scholar
  12. [12]
    R. GABASOV and F.M. KIRILLOVA, High order necessary conditions for optimality, SIAM J. Control, 10, 1972, 127–168.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    L.M. GARRIDO, General interaction picture from action principle for mechanics, J. Math. Physics., 10, 1969, 1045–1056.zbMATHCrossRefGoogle Scholar
  14. [14]
    A. ISIDORI, Nonlinear Control Systems: An Introduction, Lect. Notes Control Informat. Sci 72, Springer, Berlin, 1985.zbMATHCrossRefGoogle Scholar
  15. [15]
    D.H. JACOBSON, A new necessary condition for optimality for singular control problems, SIAM J. Control, 7, 1969, 578–595.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    H.W. KNOBLOCH, Higher Order Necessary Conditions in Optimal Control Theory, Lect. Notes Control Informat. Sci 34, Springer, Berlin, 1982.Google Scholar
  17. [17]
    A.J. KRENER, The high order maximal principle and its applications to singular extremals, Siam J. Control. Optimiz., 15, 1977, 256–293MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    F. LAMNABHI-LAGARRIGUE, Some second-order necessary conditions in optimal control, Systems Control Lett., 5, 1984, 135–143.MathSciNetCrossRefGoogle Scholar
  19. [19]
    F. LAMNABHI-LAGARRIGUE, Sur les conditions nécessaires d’optimalste du deuxième et troisième order dans les problèmes de commande optimale singulière, in “Analysis and Optimization of Systems” (A. Bensoussan and J.L. Lions eds), Lect. Notes Control. Informat. Sci 63, 525–541, Springer, Berlin, 1984.CrossRefGoogle Scholar
  20. [20]
    F. LAMNABHI-LAGARRIGUE, Série de Volterra et commande optimale singulière, Thèse d’Etat, Université Paris XI, mars 1985.Google Scholar
  21. [21]
    C. LESIAK and A.J. KRENER, The existence and uniqueness of Volterra series for nonlinear systems, IEEE Trans. Automat. Control, 23, 1978, 1090–1095.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    L. PONTRYAGIN, V. BOLTYANSKI, R. GAMKRELIDZE and E. MISCHTCHENKO, The Mathematical Theory of Optimal Processes, John Wiley, New-York, 1962, French version: Mir, Moscou, 197zbMATHGoogle Scholar
  23. [23]
    W.J. Rugh, Nonlinear System Theory, The Johns Hopkins University, Press, Baltimore, 1981.zbMATHGoogle Scholar
  24. [24]
    M. Schetzen, The Volterra and Wiener Theories of Nonlinear systems, Wiley, New York, 1980.zbMATHGoogle Scholar
  25. [25]
    R. SIEGMUND-SCHULTZE, Die Anfänge der Funtionalanalysis und ihr Platz in Umwälzungsprozessen der Mathematik um 1900, Arch. History Exact Sci, 26, 1982, 13–71.MathSciNetzbMATHGoogle Scholar
  26. [26]
    H.J. SUSSMANN, A Lie-Volterra expansion for nonlinear systems, in “Mathematical Theory of Netwoeks and Systems” (P. Fuhrmann ed.), Lect. Notes Control informat. Sci 58, 822–828, Springer, Berlin, 1984.CrossRefGoogle Scholar
  27. [27]
    C.A. UZES, Mechanical response and the initial value problem, J. Math. Physics, 19, 1978, 2232–2238.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    V. VOLTERRA, Leçons Sur Les Fonctions de Lignes, Gauthier-Villars, Paris, 1913.Google Scholar
  29. [29]
    V. VOLTERRA and J. PERES, Théorie Générale Des Fonctionnelles, 1, Gauthier-Villars, Paris, 1936.Google Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Michel Fliess
    • 1
  • Françoise Lamnabhi-Lagarrigue
    • 1
  1. 1.Laboratoire des Signaux et SystèmesC.N.R.S. — E.S.E.Gif-Sur-YvetteFrance

Personalised recommendations