In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set {(T(-)x : x £ X, \\x\\ < 1} is relatively compact in any space LP([Q, a]) ; X) for 1 < p...

In this paper, we study infinite dimensional stochastic systems having both unbounded control and observation operators. First of all, using a semigroup approach, give another take the well-p...

The existence and stability properties of a class of partial functional differential equations are investigated. The problem is formulated as an abstract ordinary functional differential equation of the form du(t)/dt = Au(t) + F(u{), where A is the infinitesimal generator of a strongly continuous semigroup of linear operators T(t), t > 0, on a Banach space X and F is a Lipschitz operator from C...

Nonlinear dynamical automata (NDAs) are implementations of Turing machines by nonlinear dynamical systems. In order to use them as parsers, the whole string to be processed has to be encoded in the initial conditions of the dynamics. This is, however, rather unnatural for modeling human language processing. I shall outline an extension of NDAs that is able to cope with that problem. The idea is...

Abstract. We consider the approximation of semigroups e and of the functions φj(τA) that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions e and φj(τA), respectively, in efficiently computable terms. This is important for semigroups, where the new approximation is different from well-known approximations by ...