Second order necessary conditions for optimal control problems of evolution equations involving final point equality constraints

We establish some second order necessary conditions for optimal control problems of evolution equations involving final point equality and inequality constraints. Compared with the existing works, main difference is due to presence end-point With such constraints, we cannot simply use variational techniques since perturbations a given may be no longer admissible. also Ekeland’s principle, which first obtain conditions. Instead, combine inverse mapping theorems on metric spaces linearization data our results.