38 research outputs found

### On measuring unboundedness of the $H^\infty$-calculus for generators of analytic semigroups

We investigate the boundedness of the $H^\infty$-calculus by estimating the
bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$:
$f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates
the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic
functions on a domain strictly containing the spectrum of $A$. We show that
$b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas
$b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result
by Vitse and complements work by Haase and Rozendaal for non-analytic
semigroups. We discuss the sharpness of our bounds and show that single square
function estimates yield
$b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$.Comment: Preprint of the final, published version. In comparison with previous
version, Prop. 2.2 was added and Thm. 3.5 has been slightly adapted in order
to point out the major assertio

### Functional calculus for $C_0$-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

### Generators with a closure relation

Assume that a block operator of the form
$\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right)$, acting
on the Banach space $X_{1}\times X_{2}$, generates a contraction
$C_{0}$-semigroup. We show that the operator $A_{S}$ defined by
$A_{S}x=A_{1}\left(\begin{smallmatrix}x\\SA_{2}x\end{smallmatrix}\right)$ with
the natural domain generates a contraction semigroup on $X_{1}$. Here, $S$ is a
boundedly invertible operator for which \epsilon\ide-S^{-1} is dissipative
for some $\epsilon>0$. With this result the existence and uniqueness of
solutions of the heat equation can be derived from the wave equation.Comment: 9 page

### On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary
control systems are continuous for arbitrary bounded input functions. This
question naturally appears when working with boundary controlled, linear
partial differential equations. Here, we focus on parabolic equations which
allow for operator-theoretic methods such as the holomorphic functional
calculus. Moreover, we investigate stronger conditions than continuity leading
to input-to-state stability with respect to Orlicz spaces. This also implies
that the notions of input-to-state stability and integral-input-to-state
stability coincide if additionally the uncontrolled equation is dissipative and
the input space is finite-dimensional.Comment: 19 pages, final version of preprint, Prop. 6 and Thm 7 have been
generalised to arbitrary Banach spaces, the assumption of boundedness of the
semigroup in Thm 10 could be droppe

### Infinite-dimensional input-to-state stability

In this talk we discuss infinite-dimensional versions of well-known stability notions relating the external input $u$ and the state $x$ of a linear system governed by the equation $\dot{x}=Ax+Bu, \quad x(0)=x_{0}.$ Here, $A$ and $B$ are unbounded operators. For instance, the system is called \textit{$L^{p}$-input-to-state stable} if $u(\cdot)\mapsto x(t)$ is bounded as a mapping from $L^{p}(0,t)$ to the state space $X$ for all $t\u3e0$. In particular, we elaborate on the relation of this notion to \textit{integral input-to-state} stability and \textit{(zero-class) admissibility} with a special focus on the case $p=\infty$.\\ This is joint work with B.~Jacob, R.~Nabiullin and J.R.~Partington