In recent papers, Loewen and Zheng, and Zeidan, introduced sets of "generalized conjugate points," say Cl(x) and Cj(x), applicable to certain optimal control problems. These sets present two undesirable features. First of all, their nonemptiness has been established merely as a sufficient condition for the existence of negative second variations. Second, one can easily find examples for which, to solve the question of nonemptiness of these sets, may be much more difficult than directly finding variations that make the second variation negative. For the fixed-endpoint problem in the calculus of variations, both difficulties are solved by means of a third set S(x) which we recently introduced. In this setting, it is a simple fact to show that Cl(x) C Cj(x) C S(x).
However, it is not known if the three sets coincide, and a comparison between them may be extremely cumbersome. In fact, there are examples for which it is straightforward to prove that S(x) ~ O, but determining the sets C1 (x) or Cj(x) may be a very difficult or perhaps even a hopeless task. In this paper, we make use of the Sturm-Liouville theory to show that, in the one-dimensional case, and under certain assumptions on the functions delimiting the problem, the three sets coincide. Q 2004 Elsevier Ltd. All rights reserved.
Keywords--Calculus of variations, Generalized conjugate points, Nonsingular extremals, Sturm-Liouville boundary-value problems.