In this note we announce preliminary results obtained by applying the methodology of Sussmann [4, 5, 6, 7, 8] to the problem studied by Ledzewicz and Schättler (cf. [1, 2]), of deriving high-order necessssary conditions for a minimum in optimal control theory, extending the classical Pontryagin Maximum Principle (abbr. MP) of [3]. The work of [1, 2] uses high-order generalizations of the theore...