Construction of the optimal outcome function for a time-optimal problem on the basis of a symmetry set

The set of the symmetry is introduced into consideration, allowing to construct the optimal outcome function for one class of time-optimal problems. Formulas for a finding of this set on a plane are offered. The results find application at studying of geometry of nonconvex sets, in the theory of optimal control and in the theory of positional differential games at studying nonsmooth features of sets of attainability, stable bridges. Besides results can be useful to experts on the equations of Hamilton–Jacobi types, working within the limits of various concepts of the generalized solutions of these equations.