Generalized Curves and Extremal Points

A nonparametric variational problem is considered in the setting of the theory of generalized curves. Instead of minimizing a functional dependent on a curve joining two given points, a functional defined on a set of Radon measures is considered ; the set of measures is determined by the boundary conditions. It is shown that this functional attains its minimum at an extremal point of the set of measures. Further, an approximation scheme is developed so that the solution of the variational problem can be effected by solving a sequence of finite-dimensional programming problems; it is possible then to construct a sequence of curves such that the functional takes along this sequence values approaching its minimum over the set of measures. It is shown that this minimum is attained at an (extremal) measure which is a generalized curve, that is, the weak${}^* $ limit of a sequence of curves. This generalized curve is characterized in terms of the minimizing elements of the sequence of discrete programming probl...