Unification in the game problem of convergence and the property of stability

The paper studies the game problem of the convergence of a conflict-controlled system with a target set in the phase space of the system at a fixed time moment, the moment the game ends. The key property of $u$-stability, introduced in the second half of the 20th century by N.N. Krasovsky and A.I. Subbotin, is investigated in the paper. The study is based on conflict-induced controlled system unification constructions, which are introduced into the game problem of convergence within the framework of the Hamilton — Jacobi formalism. The paper introduces the concepts of $u$-stable and maximal $u$-stable paths, which are dual to the concepts of $u$-stable and maximal $u$-stable bridges introduced by N.N. Krasovsky and A.I. Subbotin. The concepts of approximating systems (A-systems), i. e. systems of sets in phase space, approximating maximum $u$-stable bridge and maximum $u$-stable path in gaming the problem of convergence, are defined. In this case, the concept of the maximal $u$-stable path is an obvious analogue of the notion of a trajectory in the theory of ordinary differential equations, and the concept of an A-system for this path is an analogue of the concept of an Euler broken line. These new concepts also have features that are introduced by the presence interference (i.e., the second player) in the dynamics of a conflict-controlled system.