No abstract

A new approach for constructing vector Lyapunov function for nonlinear nonautonomous 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.

We consider an ensemble of trajectories generated by a linear differential equation subjected to disturbance and parameterized by the initial state. The scalar output of the system is the volume comprised by the states of the whole ensemble. Already the unperturbed dynamics is assumed to be unstable. In order to stabilize the system with unknown inputs in the ISS sense we design impulsive control actions based in the output signal and establish conditions under which the system possesses the ISS property under these impulsive actions.