Vector Lyapunov functions and the technique in perturbation theory

Abstract

In [l] a new comparison theorem is developed that connects the solution of perturbed and unperturbed differential systems in a manner useful in the theory of perturbations. This comparison result blends, in a sense, the two approaches, namely the method of Lyapunov functions and the method of variation of parameters, and consequently provides a flexible mechanism to preserve the nature of perturbations. The results that are given in [l] show that the usual comparison theorem in terms of a Lyapunov function is included as a special case and that perturbation theory could be studied in a more fruitful way. In the study of large scale dynamic systems [S] by the method of decomposition and aggregation, several Lyapunov functions result in a natural way. This is because one assumes that the given large scale system is decomposed into isolated subsystems and interconnections between them. If the solution of these subsystems possesses good behavior like uniform asymptotic stability, then one can construct a Lyapunov function relative to each subsystem which can then be utilized to investigate the stability properties of the given large scale dynamic system as a perturbed system by employing the method of vector Lyapunov functions. In this