Transfer functions for infinite-dimensional systems

In this paper, we study three definitions of the transfer function for an infinite-dimensional system. The first one defines the transfer function as the expression $C(sI-A)^{-1}B+D$. In the second definition, the transfer function is defined as the quotient of the Laplace transform of the output and input, with initial condition zero. In the third definition, we introduce the transfer function as the quotient of the input and output, when the input and output are exponentials. We show that these definitions always agree on the right-half plane bounded to the left by the growth bound of the underlying semigroup, but that they may differ elsewhere

An H∞ calculus of admissible operators

Abstract—Given a Hilbert space and the generator A of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any g(−s) ∈ H ∞ we show that there exists an inf nite-time admissible output operator g(A). If g is rational, then this operator is bounded, and equals the “normal” def nition of g(A). In particular, when g(s) = 1/(s + α), α

A functional analytic approach towards nonlinear dissipative well-posed systems

The aim of this paper is to develop a functional analytic approach towards nonlinear systems. For linear systems this is well known and the resulting class of well-posed and regular linear systems is well studied. Our approach is based on the theory of nonlinear semigroup and we explain it by means of an example, namely equations of quasi-hyperbolic type

Functional calculus for C0C_0-semigroups using infinite-dimensional systems theory

In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.Comment: 6 page

Stability and boundedness of continuous- and discrete-time systems

In this paper we investigate the relation between discrete- and continuous-time systems. More precisely, we investigate the stability properties of the semigroup generated by A, and the sequence $A_d^n$, $n=1,2,...$, where $A_d = (I + A)(I - A)^{-1}$

Some remarks on adaptive stabilization of infinite-dimensional systems

It is the purpose of this note to show that a first-order adaptive controller stabilizes a large class of infinite-dimensional systems described by strongly continous semigroups. It is assumed that the plant is minimum-phase and has invertible high-frequency gain. Knowledge of the sign of the high-frequency gain is not required

Stability analysis in continuous and discrete time, using the Cayley transform

For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch–Schappacher perturbations