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Marcus Waurick

Publications and source records attributed to Marcus Waurick.

82 records · Page 5Linked to original sources
arXiv2013

Boundary systems and (skew-)self-adjoint operators on infinite metric graphs

We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality preserving mappings, these characterizations can be transferred to abstract boundary value spaces of (skew-)symmetric operators. Introducing the notion of boundary systems we then present a unified treatment of different versions of boundary triples and related concepts treated in the literature. The application of the abstract results yields a description of all (skew-)self-adjoint realizations of Laplace and first derivative operators on graphs.

math.FA↗
arXiv2013

On Non-Autonomous Integro-Differential-Algebraic Evolutionary Problems

In this article, we show that a technique for showing well-posedness results for evolutionary equations in the sense of [13] established in [16] applies to a broader class of non-autonomous integro-differential-algebraic equations. Using the concept of evolutionary mappings we prove that the respective solution operators do not depend on certain parameters describing the underlying spaces in which the well-posedness results are established.

math.AP↗
arXiv2013

G-convergence of linear differential equations

We discuss $G$-convergence of linear integro-differential-algebaric equations in Hilbert spaces. We show under which assumptions it is generic for the limit equation to exhibit memory effects. Moreover, we investigate which classes of equations are closed under the process of $G$-convergence. The results have applications to the theory of homogenization. As an example we treat Maxwell's equation with the Drude-Born-Fedorov constitutive relation.

math.FA↗
arXiv2013

On a Comprehensive Class of Linear Control Problems

We discuss a class of linear control problems in a Hilbert space setting. This class encompasses such diverse systems as port-Hamiltonian systems, Maxwell's equations with boundary control or the acoustic equations with boundary control and boundary observation. The boundary control and observation acts on abstract boundary data spaces such that the only geometric constraint on the underlying domain stems from requiring a closed range constraint for the spatial operator part, a requirement which for the wave equation amounts to the validity of a Poincare-Wirtinger-type inequality. We also address the issue of conservativity of the control problems under consideration.

math.OC↗
arXiv2013

On Non-Autonomous Evolutionary Problems

The paper extends well-posedness results of a previously explored class of time-shift invariant evolutionary problems to the case of non-autonomous media. The Hilbert space setting developed for the time-shift invariant case can be utilized to obtain an elementary approach to non-autonomous equations. The results cover a large class of evolutionary equations, where well-known strategies like evolution families may be difficult to use or fail to work. We exemplify the approach with an application to a Kelvin-Voigt-type model for visco-elastic solids.

math.AP↗
arXiv2012

On a Class of Boundary Control Problems

We discuss a class of linear control problems in a Hilbert space setting, which covers diverse systems such as hyperbolic and parabolic equations with boundary control and boundary observation even including memory terms. We introduce abstract boundary data spaces in which the control and observation equations can be formulated without strong geometric constraints on the underlying domain. The results are applied to a boundary control problem for the equations of visco-elasticity.

math.OC↗
arXiv2012

A Functional Analytic Perspective to Delay Differential Equations

We generalize the solution theory for a class of delay type differential equations developed in a previous paper, dealing with the Hilbert space case, to a Banach space setting. The key idea is to consider differentiation as an operator with the whole real line as the underlying domain as a means to incorporate pre-history data. We focus our attention on the issue of causality of the differential equations as a characterizing feature of evolutionary problems and discuss various examples. The arguments mainly rely on a variant of the contraction mapping theorem and a few well-known facts from functional analysis.

math.FA↗
arXiv2012

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.

math.CA↗
arXiv2012

A note on elliptic type boundary value problems with maximal monotone relations

In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as "coefficients". A reformulation of the respective problems is constructed such that they turn out to be unitary equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of "tailor-made" distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach. DeGruyter, 2011]. The abstract framework is illustrated by various examples.

math.AP↗