Extended conjugate points in the calculus of variations

This paper concerns a characterization of second‐order optimality conditions for the fixed‐endpoint problem in the calculus of variations. The key new concept is a set S(x) with the property that S(x)=∅if and only if the second variation with respect to x, independently of non‐singularity assumptions, is non‐negative along admissible variations. We show that, for this set of points, it may be much easier (and never more difficult) to prove its non‐emptiness than directly finding variations that make the second variation negative. Earlier Loewen and Zheng, and Zeidan, introduced related sets C1 (x) and C2 (x), applicable to certain optimal control problems, whose non‐emptiness has been established merely as a sufficient condition for the existence of negative second variations. These sets, when reduced to the problem we are considering, are related according to C1 (x) ⊂C2 (x) ⊂S(x). Contrary to the behaviour of S(x), verifying membership of C1 (x) or C2 (x) may be more difficult than verifying directly if the second‐order condition holds. We provide several examples for which it is straightforward to prove that S(x) ≠∅, but determining the sets C1 (x) or C2 (x) may be a very difficult or perhaps even a hopeless task.