Estimate of the stability defect for a positional absorption set subjected to discriminant transformations

We study bordering “paths”, i.e., sets in the position space of an approach-evasion differential game that contain the positional absorption set. The positional absorption set provides an exact (classical) solution of the game. At the same time, its border is nonsmooth, which complicates the construction of this set. On the contrary, a set different from the positional absorption set may not provide an exact solution of the game but can be constructed with relative ease, for example, with the help of analytical formulas. There may be other arguments for using “paths” for solving a game. For example, the smoothness of the boundary of a chosen “path” allows one to efficiently form the players' control procedures guaranteeing the solution of a game problem in the “soft” setting by taking the motion of a conflict-controlled system to a neighborhood of the target set. In this paper, we propose a procedure for smoothing a set in a part of variables; the procedure is based on discriminant transformations. We study the stability defect caused by changing the positional absorption set of a differential game by a set-“path” with boundary that is smooth in the space variables. An estimate for the stability defect of the constructed set is presented. The results are illustrated by the example of a known differential game.