Investigation of ultimate boundedness conditions of mechanical systems via decomposition

A mechanical system with linear velocity forces and nonlinear homogeneous positional ones is studied. It is required to obtain conditions for the ultimate boundedness of motions of this system. To solve the problem, the decomposition method is used. Instead of the original system of the second order equations, it is proposed to consider two auxiliary subsystems of the first order. It should be noted that one of these subsystems is linear, and another one is homogeneous. Using the Lyapunov direct method, it is proved that if the zero solutionsof the isolated subsystems are asymptotically stable, and the order of homogeneity of the positional forces is less than one, then the motions of the original system are uniformly ultimately bounded. Next, conditions are determined under which perturbations do not disturb the ultimate boundedness of motions. A theorem on uniform ultimate boundedness by nonlinear approximation is proved. In addition, it was shown thatfor some types of nonstationary perturbations with zero mean values the conditions of this theorem could be relaxed. A mechanical system with switched nonlinear positional forces is also investigated. For the corresponding family of systems, a common Lyapunov function is constructed. The existence of such a function ensures that the motions of the considered hybrid system are uniformly ultimately bounded for any admissible switching law. Examples are presented demonstrating the effectiveness of the developed approaches.