Hamiltonian analysis of the generalized problem of Bolza

On étudie le problème généralisé de Bolza en calcul des variations. Presented at the International Conference on the Calculus of Variations held to honour the memory of Leonida Tonelli, Scuola Normale Superiore, Pisa, March 1986. On obtient des conditions nécessaires en forme hamiltonienne, sous des hypothèses moins exigeantes qu’antérieurement, en particulier sans qualification sur les contraintes. Le lien avec les problèmes de contrôle optimal est développé, ainsi que l’apport de ces conditions à la théorie de la régularité de la solution. We obtain necessary conditions in Hamiltonian form for the generalized problem of Bolza in the calculus of variations. These are proven in part by an extension to Hamiltonians of Tonelli’s method of auxiliary Lagrangians. One version of the conditions is of a new character since it is obtained in the absence of any constraint qualification on the data. A new regularity theorem is shown to be a consequence of the necessary conditions.