Optimality in infinite-horizon variational problems under sign conditions

We consider infinite-horizon variational problems on several spaces of curv es. We establish relations between these problems and the properties of the ir solutions. Notably, we exhibit situations where optimality in a given sp ace of curves implies optimality in a bigger space of curves, We work with a domain of definition of the Lagrangian which has a very general form and we provide assumptions to ensure a satisfactory theory of the necessary con ditions of optimality. We apply these results to actualized Lagrangians.