On stability of equilibria of nonlinear continuous-discrete dynamical systems

In this paper the main attention is paid to discussing the issues on sufficient conditions for Lyapunov stability of nonlinear hybrid (continuous-discrete) systems, that is, the systems, the processes in which have several levels of different descriptions and the states involve both continuous and discrete components. It is well-known that by switchings between unstable regimes in a continuous dynamical system one can achieve a stability and vice versa, even when all regimes of the continuous system are stable, under the switching there can appear unstable regimes in the system. This is why it is important to make a detailed analysis on the stability issues while passing from continuous to the hybrid system.
In the present paper we propose new tests for Lyapunov stability of stationary regimes of nonlinear hybrid system with a constant discretization step $h>0$. These tests are based on the methods for studying the stability by the linear approximation and on the formulae from the perturbation theory, which allow us to analyse the equilibria and cycles of the dynamical systems depending on a small parameter. The proposed approaches are based on a passage from the original hybrid system to equivalent in a natural sense dynamical system with a discrete time. We discuss relations between dynamical characteristics of hybrid and discrete systems. While studying the main problem on Lyapunov stability of an equilibrium of the hybrid system, we consider two formulations: the stability for small $h>0$ and stability for arbitrary fixed $h=h_{0}>0$. Moreover, we discuss some questions on scenarios of bifurcation behavior of the hybrid system under the stability loss of the equilibrium. We adduce an example illustrating the efficiency of the obtained results in the problem on studying the stability of the equilibria of the hybrid systems.