Differentiability of projections onto cones and sensitivity analysis for optimal control

The differentiability properties of the metric projections on the cones of nonnegative functions are considered. It is shown that the metric projection mapping is Bouligand differentiable in L^p(0, 1), but it is not Bouligand differentiable in W^1,p(0, 1). Using differentiability in L^p(0, 1) the application of Robinson´s implicit function theorem for nonsmooth equations to sensitivity analysis for optimal control problems is presented.