In this paper, we deal with the fixed-endpoint problem in the calculus of variations involving a delay in the phase coordinates.
Necessary conditions in the form of a maximum principle are well known and, hence, conditions equivalent to those of Euler,
Legendre, and Weierstrass. However, no results seem to exist for sufficiency, or for a corresponding Jacobi condition. We
derive necessary and
... [Show full abstract] sufficient conditions in terms of the first and second variations, extending the clssical results for
the delay-free case. This is obtained directly, that is, without referring to concepts such as conjugate points, fields of
extremals, Riccati equations, or the Hamilton-Jacobi partial differential equation. The first-order condtion is then characterized
in terms of the Euler equation, together with smoothness properties of solutions. Several examples illustrate the usefulness
of the conditions obtained.