His scientific work was developing under direct influence of Lev Semenovich Pontryagin (1908-1988), Yevgeniy Frolovich Mishchenko, Igor Rostislavovich Shafarevich.

• author of over 150 scientific publications, including 5 books,
• scientific advisor of 15 Ph.Ds and 2 Doctors of Science.

The primary scientific results of professor Zelikin belong to the area of differential equations and theory of optimal control. He has conducted outstanding research in developing optimal strategies for differential games (including game with incomplete information), in deriving optimality conditions for integrals of differential forms on curves and manifolds, in analyses of Riccati equations, in building optimal synthesis for problems of optimal control. He developed the theory of optimal synthesis, with trajectories containing a countable number of switches on a finite time interval (chattering), for multidimensional non-linear problems. These results allowed to solve a number of applied problems, such as the problem of optimal control over the rotation of a solid body, the problem of control over the motion of a material point of variable mass in Newtonian force field (the problem of space navigation), problems of robotics and mathematical economics. For his mathematical works professor Zelikin was awarded the Chebyshev Award in 1987 and the Lyapunov Award in 2010. In 1988 he defended his doctoral thesis “Methods of differential forms in the theory of optimal control”. Zelikin is a member of editorial board of “Fundamental and applied mathematics”, a member of the academic council of Peoples’ Friendship University of Russia, an officer of board Moscow Mathematical Society. Since 1992 he has been the science deputy head of the GPC department. Scientific and educational work of Zelikin for the department is exceptionally versatile. He give lectures on the basic course of the department, on several special courses, leads seminars, devoted to optimal synthesis and geometrical methods in optimal control problems.