Equivalent substitution in the control theory

In this paper we study a problem of looking for an optimal solution of a system of the differential equations with a control and an optimized function. The system of differential equations is changed for two systems with the upper and lower convex envelopes of a function on the right side of the initial differential system and the lower envelope of the optimized function in a region of attainability. The necessary conditions of optimality are sufficient for the substituted system. The rules for evaluation of the attainability set with the help of positively definite funtions are given.