Vasylieva I. Motion stabilization of mechanical control systems with random influence in critical cases

The thesis is devoted to actual problems of theoretical mechanics, which arise in the study of the stability of motion of mechanical systems described by nonlinear differential equations with random influence. A significant part of the thesis is devoted to the qualitative analysis of the behavior of mechanical systems described by stochastic differential equations. Sufficient conditions for the attraction to the invariant manifold of an arbitrary dimension in terms of the density function of a measure that has the property of monotonicity on the flow are proved. Sufficient conditions for the asymptotic stability in probability of the invariant sets of system of Ito stochastic differential equations are obtained and this result allows us to construct a stabilizing feedback control. Problems of stabilizing the motion of a rigid body around its center of mass under the action of jet control torques and a pair of flywheels are considered. We proposed the controls, which allows to ensure the partial asymptotic stability in probability and to guarantee the stochastic boundedness of trajectories of the corresponding closed-loop system. The constructive proof of Artstein's theorem generalized to the problem of partial stabilization of stochastic Ito differential equations. This result applied to nonlinear systems with stochastic effects that describe the dynamics of the inverted pendulum with moving mass and a three-wheeled trolley with an additional degree of freedom and the balancing robot. In another essential part of the thesis, the problem of stabilization of the motion of the general class of affine control systems along a given curve is considered. The concept of stability of a family of sets in case of stability concerning a part of variables is widespread. New constructive sufficient conditions for the asymptotic stability of a one-parameter family of sets using time-dependent control in the form of trigonometric polynomials are obtained.