First- and Second-Order Optimality Conditions for Optimal Control Problems of State Constrained Integral Equations

Abstract

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-order necessary conditions of optimality are given by the description of the set of Lagrange multipliers. Second-order necessary conditions are expressed by the nonnegativity of the supremum of some quadratic forms. Second-order sufficient conditions are also obtained in the case where these quadratic forms are of Legendre type.