Summary: The theory of control analyzes the proprieties of commanded systems. Problems of optimal control have been intensively investigated in the world literature for over forty years. During this period, series of fundamental results have been obtained, among which should be noted the maximum principle [

*M. Aidene, I. L. Vorobi’ev* and

*B. Oukacha*, Zh. Vychisl. Mat. Mat. Fiz. 45, No. 10, 1756–1765 (2005); translation in Comput. Math. Math. Phys. 45, No. 10, 1691–1700 (2005;

Zbl 1090.49002)] and dynamic programming [

*N. V. Balashevich, R. Gabasov* and

*F. M. Kirillova*, Comput. Math. Math. Phys. 40, No. 6, 799–819 (2000); translation from Zh. Vychisl. Mat. Mat. Fiz. 40, No. 6, 838–859 (2000;

Zbl 0990.49023)]. For many of the problems of the optimal control theory adequate solutions are found [

*R. E. Bellmann*, Dynamic programming, Princeton, NJ: Princeton University Press (1963);

*A. E. Bryson* and

*Y.-C. Ho*, Applied optimal control, Toronto, Canada: Blaisdell (1969);

*R. Gabasov, F. M. Kirillova* and

*N. V. Balashevich*, SIAM J. Control Optimization 39, No. 4, 1008–1042 (2000;

Zbl 0978.49028)]. Results of the theory were taken up in various fields of science, engineering, and economics. The present paper aims at extending the constructive methods of

*R. F. Gabasov* and

*F. M. Kirillova* [Linear programming methods. Part 1. General problems. Minsk: Izdatel’stvo Belorusskogo Universiteta (1977;

Zbl 0468.90035); Linear programming methods. Part 2. Transportation problems. Minsk: Izdatel’stvo Belorusskogo Universiteta (1978;

Zbl 0468.90036); Linear programming methods. Part 3. Special problems. Minsk: Izdatel’stvo Belorusskogo Universiteta (1980;

Zbl 0484.90067)] that were developed for the problems of optimal control with the bounded initial state is not fixed.