Topological and Invariance Entropy for Infinite-Dimensional Linear Systems

For linear (control) systems on infinite-dimensional state spaces with finite-dimensional unstable subspace, this paper introduces the concepts of topological entropy and invariance entropy. For linear dynamical systems on Banach spaces, described by a strongly continuous semigroup, the topological entropy is given by the sum of the real parts of the unstable eigenvalues of the infinitesimal generator. An application is provided by computing the topological entropy of delay equations and of a parabolic partial differential equation. Furthermore, the invariance entropy for infinite-dimensional linear control systems is equal to the topological entropy of the homogeneous equation and so it is also described by the eigenvalues of the infinitesimal generator.