Abstract. The concept of matrix resolving function is introduced to study dynamic game problems. The sufficient conditions are derived ensuring the possibility for the pursuer to bring the trajectory of a conflict-controlled process to the terminal set. The cases of using quasi-strategies and counter-controls by the pursuer are analyzed separately. Guaranteed times of the game termination for different method’s schemes are compared. The theoretical results are illustrated with a model example of “simple motions” on a plane.

Keywords: conflict-controlled process, resolving function, multi-valued mapping, Pontryagin condition, measurable choice theorem, extremal selector, H-convex set, eigenvalue, superpositional measurability, cylindrical terminal set, Aumann integral.