Asymptotic behavior of a class of abstract dynamical systems

Abstract

In recent years the extension of the second or direct method of Liapunov to distributed parameter systems has gained much attention; specifically Movchan [I] and Zubov [2] have given results paralleling Liapunov’s classical theorems. However, in dealing with the question of asymptotic stability of equilibria, these theorems are not always applicable because of their rather stringent hypotheses on the negative definiteness of the time derivative of Liapunov type functionals. In [3] Barbashin and Krasovskii did weaken somewhat the hypotheses of Liapunov’s theorems for autonomous and periodic systems but it was LaSalle [46] who exploited the invariance of limit sets to obtain a much more general theory, his principal theorem being stated in the form of an invariance principle. This was then extended by Hale [7] for the infinite dimensional space of autonomous functional differential equations and by Miller in [8,9] f or almost periodic ordinary and functional differential equations. These developments suggest the desirability of formulating invariance principles for infinite dimensional spaces suitable for partial differential equations. This paper is in that direction. The direct generalization of autonomous systems of ordinary differential equations to infinite dimensions is the abstract dynamical system. Thus, one would like to state an invariance principle similar to the theorem of LaSalle for autonomous ordinary differential equations which is valid for dynamical systems in general. Brayton and Miranker [lo] stated a type of invariance principle for abstract dynamical systems but their work was limited to a particular class of systems