Programmed Iteration Method in Differential Games with a Functional Target Set

A variant of the programmed iteration method is developed for solving game problems of realizing trajectories of a nonlinear conflict controlled system in a given set of functions. Solving these open-loop control problems yields an iterative process in a space of sets whose elements are functional positions, and the corresponding limit determines the set of successful solvability of the original problem in the class of set-valued quasi-strategies (nonanticipating responses to disturbances). For the conflict controlled system, the conditions of generalized uniqueness and uniform boundedness of programmed motions are assumed to hold. The use of an infinite-dimensional space of functional positions is essential, since the arising differential game generally does not satisfy the alternative solvability conditions in classes of feedback strategies.