An invariance principle for compact processes

The classical Liapunov theory of asymptotic stability relies on the construction of a Liapunov functional with a negative definite time derivative. Nevertheless, for a wide class of evolutionary equations of mathematical and physical interest, the inherent dissipative mechanism yields “natural” Liapunov functionals with negative semidefinite time derivatives. It was pointed out by LaSalle [l] that in the case of autonomous systems of ordinary differential equations the negative definiteness condition can be weakened if one takes into account the invariance of limit sets of so1utions.l An asymptotic stability theory which utilizes the invariance principle is now available for general topological dynamical systems (Hale [3]). In view of the success of this approach, a search was conducted for uncovering cases of nonautonomous systems of ordinary differential equations endowed with an invariance principle. Generalized forms of invariance principles were established for the classes of asymptotically autonomous (Markus [4], Opial [5]), periodic (LaSalle [6]) and asymptotically almost periodic systems (Miller [7]). Sell [8] introduces the concept of “limiting equations” and deduces the above results in a systematic way. Invariance principles have also been considered for other types of evolutionary equations which do not generate dynamical systems. In this category we mention Miller’s work [9] on almost periodic functional differential equations and a forthcoming article [IO] by Slemrod on periodic dynamical systems in Banach spaces. The conceptual similarity between results concerning a wide diversity of evolutionary equations motivates the development of an abstract unifying theory. In this work we study invariance in the framework of the theory of “processes.” A process is a direct generalization of the concept of a topological dynamical system. The transition from dynamical systems to