1,164 research outputs found
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
Characterisation of matrix entropies
The notion of matrix entropy was introduced by Tropp and Chen with the aim of
measuring the fluctuations of random matrices. It is a certain entropy
functional constructed from a representing function with prescribed properties,
and Tropp and Chen gave some examples. We give several abstract
characterisations of matrix entropies together with a sufficient condition in
terms of the second derivative of their representing function.Comment: Major revision. We found an error in the previous version that we
cannot repair. It implies that we no longer can be certain that the
sufficient condition of operator convexity of the second derivative of a
matrix entropy is also necessary. We added more abstract characterisations of
matrix entropies and improved the analysis of the concrete example
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
Forward-Invariance and Wong-Zakai Approximation for Stochastic Moving Boundary Problems
We discuss a class of stochastic second-order PDEs in one space-dimension
with an inner boundary moving according to a possibly non-linear, Stefan-type
condition. We show that proper separation of phases is attained, i.e., the
solution remains negative on one side and positive on the other side of the
moving interface, when started with the appropriate initial conditions. To
extend results from deterministic settings to the stochastic case, we establish
a Wong-Zakai type approximation. After a coordinate transformation the problems
are reformulated and analysed in terms of stochastic evolution equations on
domains of fractional powers of linear operators.Comment: 46 page
Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM
Elliptic boundary value problems which are posed on a random domain can be
mapped to a fixed, nominal domain. The randomness is thus transferred to the
diffusion matrix and the loading. While this domain mapping method is quite
efficient for theory and practice, since only a single domain discretisation is
needed, it also requires the knowledge of the domain mapping.
However, in certain applications, the random domain is only described by its
random boundary, while the quantity of interest is defined on a fixed,
deterministic subdomain. In this setting, it thus becomes necessary to compute
a random domain mapping on the whole domain, such that the domain mapping is
the identity on the fixed subdomain and maps the boundary of the chosen fixed,
nominal domain on to the random boundary.
To overcome the necessity of computing such a mapping, we therefore couple
the finite element method on the fixed subdomain with the boundary element
method on the random boundary. We verify the required regularity of the
solution with respect to the random domain mapping for the use of multilevel
quadrature, derive the coupling formulation, and show by numerical results that
the approach is feasible
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