A New Proof of the Jacobi Necessary Condition

In this paper, a new proof is given for Jacobi\´s "no-conjugate-point" necessary condition. For a certain class of linear-quadratic optimal control problems it is shown that the existence of a conjugate point in the interior of the extremal implies the exitence of control perturbations that lead to a reduction in cost. In a well-known way, through the concept of the Acessory Minimum Problem, this results in a no-conjugate-point condition for general optimal control problems. Important ideas used in this paper are adopted from Brakwell & Ho [1]. In contrast to earlier results, the new proof also applies if the coefficient functions of time associated with the Accessory Minimum Problem have any finite number of discontinuities.