Construction of vector Lyapunov function for nonlinear large-scale system with periodic subsystems

. A new approach for constructing vector Lyapunov function for nonlinear non-autonomous large-scale systems is proposed. It is assumed that independent subsystems are linear periodic systems. The components of the vector Lyapunov function are chosen as a quadratic form with a variable matrix. This matrix is an approximate solution of the Lyapunov matrix differential equation. This solution is constructed using the discretization method and the representation of the evolution operator proposed by Magnus. Sufficient conditions for the asymptotic stability of a trivial solution of a nonlinear large-scale system are established. The effectiveness of obtained results are illustrated by the example of stability investigation for coupled systems.