Average Relaxations of Extremal Problems

In this paper extremal problems that include averaging operation in constraints and objective are considered. The relaxation caused by a replacement of a problem without averaging with a problem that includes averaging is formally defined and investigated. Canonical form for nolinear programming problem with averaging is constructed and its conditions for optimality are derived. It is shown how optimality conditions for optimal control problems with various types of objectives and constraints can be derived using its averaged relaxation.