A new derivation of second-order conditions for equality control constraints

In this work we provide a new approach to the theory of second-order necessary conditions in optimal control, considering a problem posed over piecewise continuous controls and involving equality constrains in the controls. The proof that the second-order conditions obtained are necessary for optimality is simpler than others available in the literature and conveys a clear understanding of how to define a set of differentially admissible variations where a certain quadratic form is nonnegative.