Conjugate intervals in optimal control

This paper concerns a characterization of second-order optimality conditions for certain classes of optimal control problems. The key new concept is a set S(x,u) with the property that S(x,u)=φ if and only if the second variation with respect to (x,u) is nonnegative along admissible variations. Since this holds independently of nonsingularity assumptions, this new notion extends that of conjugacy not only in optimal control but also in the classical theory of calculus of variations. We show that, for this set of points, it may be much easier (and never more difficult) to prove its nonemptiness than directly finding variations that make the second variation negative. Earlier Loewen and Zheng, and Zeidan, introduced related sets C₁(x,u) and C₂(x,u) whose nonemptiness has been established merely as a sufficient condition for the existence of negative second variations. The three sets are related according to C₁(x,u)⊂C₂(x,u)⊂S(x,u). Contrary to the behaviour of S(x,u), verifying membership of C₁(x,u) or C₂(x,u) may be more difficult than verifying directly if the second-order condition holds. We provide simple examples for which it is straightforward to prove that S(x,u)≠φ, but determining the sets C₁(x,u) or C₂(x,u) may be a very difficult or perhaps even a hopeless task.