393 research outputs found

### 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

### Root locii for systems defined on Hilbert spaces

The root locus is an important tool for analysing the stability and time
constants of linear finite-dimensional systems as a parameter, often the gain,
is varied. However, many systems are modelled by partial differential equations
or delay equations. These systems evolve on an infinite-dimensional space and
their transfer functions are not rational. In this paper a rigorous definition
of the root locus for infinite-dimensional systems is given and it is shown
that the root locus is well-defined for a large class of infinite-dimensional
systems. As for finite-dimensional systems, any limit point of a branch of the
root locus is a zero. However, the asymptotic behaviour can be quite different
from that for finite-dimensional systems. This point is illustrated with a
number of examples. It is shown that the familiar pole-zero interlacing
property for collocated systems with a Hermitian state matrix extends to
infinite-dimensional systems with self-adjoint generator. This interlacing
property is also shown to hold for collocated systems with a skew-adjoint
generator

### $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

Hyperbolic partial differential equations on a one-dimensional spatial domain
are studied. This class of systems includes models of beams and waves as well
as the transport equation and networks of non-homogeneous transmission lines.
The main result of this paper is a simple test for $C_0$-semigroup generation
in terms of the boundary conditions. The result is illustrated with several
examples

### Variational principles for self-adjoint operator functions arising from second-order systems

Variational principles are proved for self-adjoint operator functions arising
from variational evolution equations of the form $\langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0
[z(t),y] = 0.$ Here $\mathfrak{a}_0$ and $\mathfrak{d}$ are densely defined,
symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate
with the variational evolution equation an equivalent Cauchy problem
corresponding to a block operator matrix $\mathcal{A}$, the forms $\mathfrak{t}(\lambda)[x,y] := \lambda^2\langle x,y\rangle +
\lambda\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y],$ where $\lambda\in \mathbb C$
and $x,y$ are in the domain of the form $\mathfrak{a}_0$, and a corresponding
operator family $T(\lambda)$. Using form methods we define a generalized
Rayleigh functional and characterize the eigenvalues above the essential
spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The
obtained results are illustrated with a damped beam equation.Comment: to appear in Operators and Matrice

### 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

### Analyticity and Riesz basis property of semigroups associated to damped vibrations

Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract
Hilbert space are considered. Such equations are often used as a model for
transverse motions of thin beams in the presence of damping. We derive various
properties of the operator matrix $A$ associated with the second order problem
above. We develop sufficient conditions for analyticity of the associated
semigroup and for the existence of a Riesz basis consisting of eigenvectors and
associated vectors of $A$ in the phase space

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