535 research outputs found
On measuring unboundedness of the -calculus for generators of analytic semigroups
We investigate the boundedness of the -calculus by estimating the
bound of the mapping :
for near zero. Here, generates
the analytic semigroup and is the space of bounded analytic
functions on a domain strictly containing the spectrum of . We show that
in general, whereas
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
.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 -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
, acting
on the Banach space , generates a contraction
-semigroup. We show that the operator defined by
with
the natural domain generates a contraction semigroup on . Here, is a
boundedly invertible operator for which \epsilon\ide-S^{-1} is dissipative
for some . 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
Funnel control for a moving water tank
We study tracking control for a moving water tank system, which is modelled
using the Saint-Venant equations. The output is given by the position of the
tank and the control input is the force acting on it. For a given reference
signal, the objective is to achieve that the tracking error evolves within a
prespecified performance funnel. Exploiting recent results in funnel control we
show that it suffices to show that the operator associated with the internal
dynamics of the system is causal, locally Lipschitz continuous and maps bounded
functions to bounded functions. To show these properties we consider the
linearized Saint-Venant equations in an abstract framework and show that it
corresponds to a regular well-posed linear system, where the inverse Laplace
transform of the transfer function defines a measure with bounded total
variation.Comment: 11 page
Input-to-state stability of unbounded bilinear control systems
We study input-to-state stability of bilinear control systems with possibly
unbounded control operators. Natural sufficient conditions for integral
input-to-state stability are given. The obtained results are applied to a
bilinearly controlled Fokker-Planck equation.Comment: 20 pages, completely new version based on the few preliminary ideas
in v1. Compared to v1, the results have been significantly generalized and
extende
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