  For an optimal control problem, the concept of a strong local infimum is
introduce, for which necessary conditions consisting of some family of "maximum
principles" are formulated. If a function delivers a strong local minimum in
this problem (and therefore, a~strong local infimum), then this family contains
the classical Pontryagin maximum principle. As a corollary, we derive
generalized necessary conditions for a strong local minimum for a problem of
the calculus of variations. Examples are given to show that the necessary
conditions obtained in the present paper generalize and strengthen classical
results.
