Sufficiency by a direct method in the variable state problem of calculus of variations: singular extremals

Abstract

A direct sufficiency proof for singular and non-singular extremals for the parametric variable state problem of Bolza in the calculus of variations is presented. This technique is self-contained in the sense that it makes no use of the classical concepts of conjugate or focal points, fields of extremals or certain matrix Riccati inequalities. Moreover, we also study the non-parametric variable state problem of Bolza and derive sufficient conditions for singular and non-singular extremals for weak and strong local minima.