Second-order optimality conditions for the Bolza problem with path constraints

A set of sufficient conditions for a weak minimum is derived for a form of the nonsingular Bolza problem of variational calculus, with interior point constraints and discontinuities in the system equations. Generalized versions of the conjugate point/focal point, normality, convexity and nontangency conditions associated with the ordinary Bolza problem are obtained. The resulting set of sufficient conditions is minimal, in that only minor modifications are required in order to obtain necessary conditions for normal, nonsingular problems of this form. These conditions are relatively easy to implement. Analogous second-order optimality conditions for problems with natural corners or control constraints are also obtained. Previously stated sufficiency conditions for problems with control constraints are shown to be unnecessarily restrictive, in some cases.