1,902 research outputs found
Exact observability, square functions and spectral theory
In the first part of this article we introduce the notion of a
backward-forward conditioning (BFC) system that generalises the notion of
zero-class admissibiliy introduced in [Xu,Liu,Yung]. We can show that unless
the spectum contains a halfplane, the BFC property occurs only in siutations
where the underlying semigroup extends to a group. In a second part we present
a sufficient condition for exact observability in Banach spaces that is
designed for infinite-dimensional output spaces and general strongly continuous
semigroups. To obtain this we make use of certain weighted square function
estimates. Specialising to the Hilbert space situation we obtain a result for
contraction semigroups without an analyticity condition on the semigroup.Comment: 17 page
Controllability cost of conservative systems: resolvent condition and transmutation
This article concerns the exact controllability of unitary groups on Hilbert
spaces with unbounded control operator. It provides a necessary and sufficient
condition not involving time which blends a resolvent estimate and an
observability inequality. By the transmutation of controls in some time L for
the corresponding second order conservative system, it is proved that the cost
of controls in time T for the unitary group grows at most like \exp(\alpha
L^{2}/T) as T tends to 0. In the application to the cost of fast controls for
the Schr{\"o}dinger equation, L is the length of the longest ray of geometric
optics which does not intersect the control region. This article also provides
observability resolvent estimates implying fast smoothing effect
controllability at low cost, and underscores that the controllability cost of a
system is not changed by taking its tensor product with a conservative system.Comment: 20 pages, a4paper, typos corrected in lem.5.2, lem.5.3, th.10.
How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability
Published versio
Exact observability and controllability for linear neutral type systems
The problem of exact observability is analyzed for a wide class of neutral
type systems by an infinite dimensional approach. The duality with the exact
controllabil-ity problem is the main tool. It is based on an explicit
expression of a neutral type system which corresponding to the abstract adjoint
system. A nontrivial relation is obtained between the initial neutral system
and the system obtained via the adjoint abstract state operator. The
characterization of the duality between controllability and observability is
deduced, and then observability conditions are obtained.Comment: Accepted in Systems and Control Letter
Grushin problems and control theory: Formulation and examples
In this paper we give a new formulation of an abstract control problem in
terms of a Grushin problem, so that we will reformulate all notions of
controllability, observability and stability in a new form that gives readers
an easy interpretation of these notions
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