problem, it is necessary to minimise a function of the [formula] type defined on a set of fixed trajectory-control [formula] couples, each torque satisfying the relationship [formula]. Cesari [1] has dealt with this minimisation problem in the event that (-°) is a continuous vector application. Goodman [2] provided a way to lift the hypothesis of continuity and to work with Caratheodory functions. It resolved one of the key problems in theory, namely the existence of a measurable control or (t) associated with the function x (t) itself borderline in the Cesari sense of a series of trajectories. [Formula] What we propose to do is to extend the results of L. Cesari to cases where (-°) is an application of Caratheodory, for Pontryagin’s problems, i.e. problems where controls take their values into a compact. In order to make the presentation clearer, we will partly repeat the demonstrations of L. Cesari and G. Goodman. Vergison E. On a Pontryagin problem. In: Science Classe Bulletin, Volume 56, 1970. pp. 492-504.