To global asymptotic stability of dynamical systems.

Abstract

We study the global asymptotic stability of dynamical systems with a countable state of equilibrium for two cases: 1) when the value of the integral of a periodic function in a period is equal to zero; 2) when the value of the integral is not equal to zero. A method for selecting an area of the global asymptotic stability in the space of the design parameters of the system is developed. The effectiveness of the method is demonstrated by two examples: the problem of phase synchronization; the motion of a simple pendulum. The proposed method of the study allows highlight a wider area of stability in the parameter space of the system, rather than the known methods. A distinctive feature of the proposed method by known methods (frequency, periodic Lyapunov functions) is that the conditions for the global asymptotic stability follows from the estimates of improper integrals along the solutions of the system. In the work the following results are obtained: the equations of motion of the system with the help of a smooth transformation is given to a special form; the identity along the solutions of the system and evaluation of solutions of the system; The asymptotic properties of functions with the limitations of the improper integral are studied; based on the evaluation of improper integrals along the solutions of the system, theorems on the global asymptotic stability of stationary set of dynamic systems are proved.