Stability analysis of non-stationary mechanical systems with discontinuous right-hand sides

The paper investigates the stability problem for a class of non-stationary mechanical systems under the action of linear dissipative and nonlinear potential forces. It is assumed that the system has a changeable structure. Switching between different operating modes is associated with a change of the potential of the system, as well as with discontinuities of non-stationary coefficients present in the system. Two approaches to the analysis of the stability of such systems are considered. One is related to the construction of a discontinuous Lyapunov function, the other is based on the construction of a continuous Lyapunov function. The paper also studies the effect of non-stationary perturbed forces on the stability. The peculiarity of the work is that the non-stationary parameters both in the system itself and in the perturbations can be unbounded with respect to time, or, on the contrary, they can arbitrarily approach to zero. Thus, the problem arises of comparing the rate of growth or decrease of all these non-stationarities in order to obtain conditions that guarantee the asymptotic stability of the given equilibrium position of the system.