On stability and stabilization of the second-order nonlinear equation

The article presents the solution of the problem about sufficient conditions for asymptotic stability of equilibrium position for special-kind ordinary and stochastic differential equations. Theorems obtained in the paper are applied for solution of the stabilization problem for two-dimensional rotational motion of a satellite on elliptic orbit. This motion may be influenced by random forces; parameters of the motion also may vary stochastically. Authors prove the theorem about the sufficient conditions of asymptotic stability. These conditions are based on Lyapunov function with sign-constant derivative by virtue of the ordinary differential equation and the corresponding operator by virtue of the stochastic differential equation. Novelty of the results is that new robust stability conditions are obtained. In particular the authors solved the problem about stabilization of satellite’s motion wherein it makes three turns in absolute space during a time equal to two periods of revolution of the center of mass on the orbit.