In this paper we present an integral-invariance principle generalizing LaSalle’s invariance principle for nonlinear systems, The principal new ingredients are the use of observation functions and certain integrability conditions, which are particularly well suited for dynamical systems involving control and observations, Indeed, the integral-invariance principle leads to the development of a series of results relating stability, observability, and the converse theorems of Lyapunov theory, Corollaries include apparently diverse stabilizability results for adaptive control, for nonlinear control, and for passive circuits and systems.