RUDN Journal of Engineering ResearchRUDN Journal of Engineering Research2312-81432312-8151Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)1863110.22363/2312-8143-2018-19-1-67-79Research ArticleOptimal control problem and its solution by grey wolf optimizer algorithmDiveevAskhat I<p>Doctor of Technical Sciences, professor, head of sector of Cybernetic problems, Federal Research Centre “Computer Science and Control” of Russian Academy of Sciences, professor at Department of Mechanics and Mechatronics, Engineering Academy, Peoples’ Friendship University of Russia (RUDN University). Research interests: Computational methods for problems of control</p>aidiveev@mail.ruKonstantinovSergey V<p>senior lecturer at Department of Mechanics and Mechatronics, Engineering Academy, Peoples’ Friendship University of Russia (RUDN University). Research interests: Optimization algorithms, evolutionary algorithms, genetic algorithms, computational methods for problems of optimal control</p>konstantinov_sv@rudn.universityInstitution of Russian Academy of Sciences, Dorodnicyn Computing Centre of RASPeoples’ Friendship University of Russia (RUDN University)15122018191677903062018Copyright © 2018, Diveev A.I., Konstantinov S.V.2018<p>The paper is devoted to a numerical method for solving the optimal control problem. The main approach to the numerical solution of the optimal control problem is the reduction of the optimal control problem to the problem of nonlinear programming and its following solution by classical gradient optimization methods. For this purpose, optimal control problem, which is a problem of searching time-dependent function, is replaced by the problem of searching of control values at discrete instants of time. An increase in the number of sampling points increases the accuracy of function approximation, but at the same time increases the dimensionality of the search space in the non-linear programming problem. In complex problems of non-linear programming with an unknown topology of the objective function, the statement that using classical gradient methods ensures finding a solution is not justified. The optimal control problem after the discretization and other modifications is often transformed to a non-linear programming problem with a non-unimodal objective function for which gradient methods are not applicable. In this paper we propose to solve the optimal control problem by evolutionary algorithms that do not use gradients and are able to find solutions of problems with nonunimodal objective function. The paper presents the modern evolutionary algorithm Grey wolf optimizer. The problem of the optimal combat turn of the aircraft is considered. In this problem the mathematical model of the control object is described by a system of seven ordinary differential equations. Also constraints on the value and rate of change of control are given. It is experimentally shown that the evolutionary algorithm Grey wolf optimizer successfully solves this optimal control problem.</p>optimal control problemevolutionary algorithmsgrey wolf optimizerэволюционный алгоритмоптимальное управлениеалгоритм «серого волка»[Evtushenko Yu.G. Optimizaciya i bystroe avtomaticheskoe differencirovaniye [Optimization and fast automatic differentiation]. Moscow: Dorodnicyn Computing Centre of RAS, 2013. (In Russ.).][Karpenko A.P. Sovremennyye algoritmy poiskovoi optimizacii. Algoritmy, vdohnovlennye prirodoi [Modern algorithms of search optimization. Nature-inspired algorithms]. Moscow: Bauman Press. 2014. (In Russ.).][Diveev A.I., Konstantinov S.V. Evolutionary algorithms for the problem of optimal control. RUDN Journal of Engineering Researches. 2017. Vol. 18. No. 2. Pp. 254—265. (in Russ.)][Diveev A.I., Konstantinov S.V. Study of evolutionary algorithms for the optimal control problem. Proceedings of MIPT. 2017. Vol. 9. No. 3. Pp. 76—85. (in Russ.)][Mirjalili, S., Mirjalili, S.M., Lewis, A. Grey Wolf Optimizer / In Advances in Engineering Software, 2014. Vol. 69, Pp. 46–61. DOI: 10.1016/j.advengsoft.2013.12.007.][Grachev N.I., Evtushenko Yu.G. A library of programs for solving optimal control problems, U.S.S.R. Comput. Maths. Math. Phys. 1979. Vol. 19. No. 2. Pp. 367—387. (In Russ).]