Some problems of target approach for nonlinear control system at a fixed time moment

The game problem of target approach for nonlinear control system with a target set in a finite-dimensional phase space at a fixed time is studied. The problem is formulated and studied within the framework of concepts and constructions of the theory of positional differential games, created by N.N. Krasovskii and A.I. Subbotin in the second half of the 20th century. One of the central problems of the theory of positional differential games is the problem of calculating positional sets of uptake in convergence game problems. The paper investigates a key stability property in the theory of positional differential games, which is a characteristic of some closed sets in the space of positions of a controlled system that are convenient for the first player to conduct the game. It is important that this property is also characteristic for solvability sets in convergence problems: the inclusion of the concept of stability in research makes it possible to obtain analytical descriptions of solvability sets in some specific convergence problems and to develop algorithms for approximate calculation of the solution in a number of specific problems. Some modifications of the definition of a $u$-stable bridge in the considered convergence problem and a system of sets approximating the attainability set are given. Three specific problems on the convergence of mechanical systems are also presented, computer solutions are simulated, and graphical simulation results are presented.