On a technique of stability analysis of a discrete system (Q2754754)

The author considers a discrete system \(S:\;x(\tau+1) = f[\tau, x(\tau)]\). It is assumed that the system can be decomposed into \(s\) interconnected systems \(\widetilde{S}_i: x_i(\tau+1) = g_i[\tau, x_i(\tau)] + h_i[\tau, x(\tau)]\). Each of equations describing the dynamics of independent subsystems \(S_i: x_i(\tau+1) = g_i[\tau, x_i(\tau)]\) is decomposed into \(m_i\) interconnected subsystems. To study the stability of \(S\), the author uses a two-level construction of Lyapunov function. A theorem concerning the asymptotic stability of the system and its asymptotic stability in large is proved, and some particular examples are considered.