Strong and Weak Augmentability in Calculus of Variations

In this paper we derive sufficient conditions for local optimality, for the Lagrange problem in the calculus of variations involving mixed equality constraints, by means of the notion of augmented Lagrangians. It is well-known that the standard necessary conditions for that problem can be easily obtained under the assumption of augmentability, instead of the usual one of normality. On the other hand, as we show in this paper, the standard sufficient conditions for a strong (weak) minimum imply strong (weak) augmentability. Since this kind of augmentability implies that the extremal under consideration is a local solution, the results given provide an alternative approach to the classical theory of sufficient conditions. Key–Words: Augmentability, Lagrange problem, calculus of variations, equality constraints, sufficient conditions