Conjugacy for normal or abnormal linear Bolza problems

AbstractFor certain Bolza problems with linear dynamics, two sets extending the notion of conjugate points in the calculus of variations are introduced. Independently of nonsingularity assumptions, their emptiness, in one case without normality assumptions, is shown to be equivalent to a second order necessary condition. A comparison with other notions available in the literature is given

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

AbstractIn 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