On the dynamic stability of impulsive mechanical systems with delay (Q2202628)

The asymptotic stability is studied for mechanical systems, taking into account the delays of feedback, an impulsive action at fixed times, and periodic variations of the parameters of the system (parametric perturbations). The first half of the paper presents sufficient conditions for asymptotic stability of the equilibrium for a linear system of impulsive differential equations with delay and with periodic parameters. The value of delay is is assumed to be equal to the period of the system. To study the problem, the Poincare mapping is used, so the original problem is reduced to the localization of the roots of a certain transcendental characteristic function in a unit disk of the complex plane. The practical application of the result is quite complicated, but the authors formulate less sharp but easily verifiable sufficient conditions and several corollaries for different special cases. In the second half of the paper, the conditions of parametric stability of the lower equilibrium of a mathematical pendulum are deduced with regard for the impulsive perturbations and delay. The domains of asymptotic stability in the space of two parameters are plotted for a numerical example at different values of the period. The bibliography contains, among other things, Russian-language sources, the introduction provides a fairly detailed overview of the history of the issue.