On the invariance principle: generalizations and applications to synchronization

In many engineering and physics problems it is very hard to find a Lyapunov function satisfying the classical version of the LaSalle´s invariance principle. In this work, an extension of the invariance principle, which includes cases where the derivative of the Lyapunov function along the solutions is positive on a bounded set, is given. As a consequence, a larger class of problems may now be considered. The results are used to obtain estimates of attractors which are independent of coupling parameters. They are also applied to study the synchronization of coupled systems, such as coupled power systems and coupled Lorenz systems. Estimates on the coupling term are obtained in order to accomplish the synchronization