Construction of solvability set for differential games with simple motion and non-convex terminal set

Two-players zero-sum differential games in the plane with simple motions, fixed terminal time and polygonal terminal set are considered. The geometrical constraint on control of each player is supposed to be a convex polygonal set or a linear segment. An explicit formula that describes the solvability set (the level set of the value function, maximal $u$-stable bridge, viability set) is well-known for a convex terminal set. The corresponding algorithm of construction is based on the set operations of algebraic sum and geometrical difference (Minkowski difference). We suggest an algorithm of exact construction of the solvability set in the case of non-convex terminal set. One does not need any additional partitions of the time interval and restoring intermediate sets at the additional instants. The algorithm is based on selecting and finite recursive processing a list of half-planes in three-dimensional space of time and state coordinates. The list is constructed on the basis of polygonal terminal set with usage the outer normals to the polygonal constraints of the players' controls. Some illustrative examples are computed.