Distance estimates to feasible controls for systems with final point constraints and second order necessary optimality conditions

Abstract

Abstract We prove an inverse mapping theorem on a metric space of controls that allows to “control” final points of trajectories of a nonlinear system. More precisely, our result provides local distance estimates of arbitrary controls from feasible ones. As an application we derive second-order necessary optimality conditions for L 1 -local minima for the Mayer optimal control problem with a general control constraint U ⊂ R m , state constraints described by inequalities and final point constraints, possibly having empty interior. Thanks to this inverse mapping theorem we first get a second-order variational inequality as a necessary optimality condition. Then the separation theorem leads in a straightforward way to second-order necessary conditions.