Controllability and Observability in Banach Space with Bounded Operators

Abstract

The classical theory of (state and output) controllability and observability in finite-dimensional spaces is extended to linear abstract systems defined on infinite-dimensional Banach spaces, under the basic assumption that the operator acting on the state be bounded. Tests for approximate controllability as well as observability, expressed only in terms of the coefficients of the system, are proved via a consequence of the Hahn–Banach theorem, and new phenomena arising in infinite dimensions are studied : for instance, by using Baire category arguments, it is shown that state exact controllability, under large conditions met in cases of physical interest, never arises in infinite-dimensional Banach spaces, even with free final instant. Several examples are presented throughout ; in particular, for dynamical systems modeled by integro-differential equations of Volterra type, the present theory leads in turn to explicit, easy-to-check criteria for approximate controllability and observability. An example s...