Jacobi's condition for singular extremals: An extended notion of conjugate points

For the simple fixed endpoint problem in the calculus of variations, Jacobi's condition (“there are no conjugate points in the interior of the underlying time interval”) is necessary for optimality if the trajectory under consideration is nonsingular. In this paper, we extend the notion of conjugate points so that the above condition (in terms of this new notion) is necessary also for singular extremals. This is achieved by showing that, without any additional assumption on the trajectory, the nonnegativity of the second variation on the space of admissible variations is equivalent to the nonexistence of these “extended conjugate points”.