Journal of Optimization Theory and Applications

, Volume 183, Issue 3, pp 792–812 | Cite as

A Generalization of Michel’s Result on the Pontryagin Maximum Principle

  • Joël BlotEmail author
  • Hasan Yilmaz


We provide an improvement of the maximum principle of Pontryagin of the optimal control problems, for a system governed by an ordinary differential equation, in the presence of final constraints, in the setting of the piecewise continuously differentiable state functions (valued in a Banach space) and of piecewise continuous control functions (valued in a metric space). As Michel, we use the needlelike variations, but we introduce tools of functional analysis and a recent multiplier rule of the static optimization to make our proofs.


Pontryagin maximum principle Piecewise continuous functions Fixed point theorem 

Mathematics Subject Classification

49K15 47H10 



The authors thank very much the anonymous referee who has helped us to improve the presentation of the paper.


  1. 1.
    Michel, P.: Une démonstration élémentaire du principe du maximum de Pontriaguine. Bull. Math. Écon. 14, 8–23 (1977)Google Scholar
  2. 2.
    Blot, J.: On the multiplier rules. Optimization 65, 947–955 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alexeev, V.M., Tihomirov, V.M., Fomin, S.V.: Commande Optimale, French edn. MIR, Moscow (1982)Google Scholar
  4. 4.
    Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems, English edn. North-Holland Pub. Co., Amsterdam (1979)zbMATHGoogle Scholar
  5. 5.
    Carlson, D.A.: An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation. J. Optim. Theory Appl. 54, 43–61 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clarke, F.H., Ledyaev, F.H., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)zbMATHGoogle Scholar
  7. 7.
    Bourbaki, N.: Fonctions d’une Variable Réelle. Théorie Élémentaire, Hermann (1976)zbMATHGoogle Scholar
  8. 8.
    Schwartz, L.: Cours d’Analyse. Tome 1, Hermann (1967)Google Scholar
  9. 9.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: Théorie Mathématique des Processus Optimaux, French edn. MIR, Moscow (1974)Google Scholar
  10. 10.
    Dugundji, J., Granas, A.: Fixed Point Theory, vol. 1. PWN-Polish Scientific Publishers, Warsawa (1982)zbMATHGoogle Scholar
  11. 11.
    de la Barrière, R.P.: Cours d’Automatique Théorique. Dunod, Paris (1965)zbMATHGoogle Scholar
  12. 12.
    Aubin, J.-P.: Applied Functional Analysis. Wiley, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Michel, P.: Problèmes des inégalités et application à la programmation dans le cas où l’espace d’arrivée est de dimension finie. C.R. Acad. Sci. 273(B), 389–391 (1974)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire SAMM EA 4543, Centre P.M.F.Université Paris 1 - Panthéon SorbonneParisFrance
  2. 2.Laboratoire LPSM UMR 8001Université Paris DiderotParisFrance

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