New conditions for stability of a nonautonomous discrete system (Q2754778)

A discrete system of the type \(S:\;x(\tau+1) = A(\tau)x(\tau) + h(\tau,x(\tau))\) is considered. It is assumed that the system is decomposed into \(s\) interconnected systems \(\tilde{S}_i:\;x_i(\tau+1) = A_i(\tau)x_i(\tau) + \sum_{j=1, j\neq i}^s B^{(i)}_j(\tau)x_j(\tau) + h_i(\tau,x(\tau))\). Each of equations describing dynamics of independent subsystems of this system \(S_i:\;x_i(\tau+1) = A_i(\tau)x_i(\tau)\) is decomposed into \(m_i\) interconnected subsystems. For study of stability of the system \(S\), a two-level construction of the Lyapunov function is used. A theorem concerning asymptotic stability of the system in large is proved. Several examples are considered.