Optimal control of nonlinear Fredholm integral equations

Optimal control problems with nonlinear equations usually do not possess optimal solutions, so that their natural (i.e., continuous) extension (relaxation) must be done. The relaxed problem may also serve to derive first-order necessary optimality condition in the form of the Pontryagin maximum principle. This is done here for nonlinear Fredholm integral equations and problems coercive in an L-p-space of controls with p<+infinity. Results about a continuous extension of the Uryson operator play a key role.