First and second order optimality conditions for optimal control problems of state constrained integral equations

This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions

About the relation between some optimality conditions

summary:The relation between the general optimality conditions in terms of contact cones and the Kuhn-Tucker conditions in the special case of pseudo-convex and quasi-convex functions and their consequence to Lagrangian multipliers are given