NECESSARY AND SUFFICIENT CONDITIONS LIMIT CYCLES OF THE FIRST KIND
Mathematical model of dynamic systems with cylindrical phase space is the class of ordinary differential equations, the right side which contains periodic functions of the phase coordinates of the system, called corner. Phase systems have the following characteristics : First, they have a countable set of equilibria, and secondly, in such systems, apart from the usual limit cycles ( the first kind ) can be the limit cycles of the second kind associated with the periodicity of the right-hand side of the differential equation for the angular coordinates; third, these systems have a particular form of motion – circular motion generated by preserving the sign of the angular coordinate. The problems of handling and optimum control phase systems with fixed ends of the trajectories are new developments in the theory of phase systems. In the periodic motions are considered as solutions of boundary value problems for phase systems. By introducing artificial control actions of these boundary value problems are immersed in the objectives of control. Next, on the basis of established authors of the theory of boundary value problems of controllability of control are reduced to the corresponding optimal control problems with the free right ends of the trajectories and the specific functional connected with the peculiarities of boundary value problems [1]-[5]. To obtain the necessary and sufficient condition for the existence of limit cycles prove several lemmas provide estimates of improper integrals along solutions of the phase system.