Construction of solutions in certain differential games with phase constraints

A differential approach-evasion game with fixed termination time is studied. It is assumed that the phase vector of the conflict-control system is subjected to constraints that form a closed set in the position space. The ideology of stable bridges is used for solving the problem.
A method of convolution is proposed, which is used in several problems for constructing explicitly the stable absorption operator defining the stable bridges. A method of approximate construction of the maximal stable bridge in this game is suggested. The relations are written down that define a system of sets approximating the maximal stable bridge, and a control procedure with a guide is described, which can be used for obtaining an approximate solution of the approach problem.