Vorträge

In this talk we study approximation techniques for input-output systems, which appear in the modeling process of mechanical systems. So, the focus will be on linear dynamical systems with a second derivative term.
These system can become very large in practice and therefore, expensive to be used for simulations and controller design.
Since this frequently happens to all control systems coming from real-live application, model order reduction became a major field in control theory over the last decades.
Here however, beside approximating the input-output behavior of the original system, the special structure should be preserved in the reduced-order model.
So far, reduction techniques designed for the linearized model fail in this aspect. On the other hand, there is a wide variety of methods that directly treat the second order control system. However, up to this point none of those methods deliver reasonable error-bounds for the approximation.
In this talk an approximation method is presented for the special class of passive mechanical systems. Roughly speaking passivity for control systems means that the system itself cannot produce energy. For this class the special canonical structure, given by so called Jordan triples for matrix polynomials, can be exploited.
In the end an error bound in the gap metric will be derived. The gap metric is used as a measure for the distance of two linear systems. It is defined via the distance of the closed subspaces of stable trajectories corresponding to zero initial conditions of the systems. Hence, the gap metric error-bound ensures the quality of the approximation of the state/signal system.

Abstract
We study two standard biased (1 : b) Maker-Breaker positional games
| the Perfect Matching game and the Hamilton Cycle game, played on
the edge set of the complete graph on n vertices, Kn. Given Breaker's bias
b, possibly depending on n, our goal is to determine the minimal number
of moves in which Maker can win in each of these two graph games.

This is joint work with Miloš Stojakovic.

In this talk we outline classical connections between such mathematical objects as operator semigroups, evolution equations and Markov processes. Further, we present a method to approximate operator semigroups with the help of the Chernoff theorem. Many \emph{Chernoff approximations} lead to representations of solutions of (corresponding) evolution equations in the form of limits of $n$-fold iterated integrals of elementary functions when $n$ tends to infinity. Such representations are called \emph{Feynman formulae}. They can be used for direct computations, modelling of the related dynamics, simulation of underlying stochastic processes.
In some cases, Chernoff approximations can be understood as a version of the operator splitting method (known in the numerics of PDEs); some Feynman formulae provide Euler--Maruyama schemes for SDEs. Moreover, the limits in Feynman formulae sometimes coincide with path integrals with respect to probability measures (\emph{Feynman-Kac formulae}) or with respect to Feynman type pseudomeasures (\emph{Feynman path integrals}). It is planned to discuss different Chernoff approximations for semigroups corresponding to some Markov processes (e.g., subordinate Feller diffusions on star graphs and Riemannian manifolds) and for Schr\''{o}dinger groups.
Furthermore, the constructed Chernoff approximations for operator semigroups can be used to approximate solutions of some time-fractional evolution equations describing anomalous diffusion (solutions of such equations do not posess the semigroup property).

The original problem of counting domino towers was first studied by G. Viennot in 1985, see also D. Zeilberger (The Amazing 3^n Theorem). We analyse a generalisation of domino towers that was proposed by T. M. Brown (J. Integer Seq. 20.3 (2017), Art. 17.3.1), which we call S-omino towers. After establishing an equation that the generating function must satisfy and applying the Lagrange Inversion Formula, we find a closed formula for the number of towers.

The talk should hopefully also be accessible to people not used to this kind of mathematics.

This talk is divided into two parts. The first part will be given on Thursday 08.11.18 by Dennis Gallaun.
In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an obserbability estimate with explicite dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp(Rd) and on Lp(Td) for 1 < p < ∞. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.

This talk is divided into two parts. The second part will be given on Thursday 15.11.18 by Christian Seifert.
In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an obserbability estimate with explicite dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp(Rd) and on Lp(Td) for 1 < p < ∞. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.

My talk will consist of three short, independent parts, the first one being a quick introduction of myself. In the second and the third part, I will give an ''easy-to-digest'' survey of my graduate theses [1,2].

References:

[1] Implementation and Performance Analyses of a Highly Efficient Algorithm for Pressure-Velocity Coupling. Master Thesis Computational Engineering, Darmstadt, 2015

[2] On the L^p Theory of the Stokes Operator in Lipschitz Domains. Master Thesis Mathematics, Darmstadt, 2018

Vortrag (PDF, 2.1MB)

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation.
This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over
$\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space
over a field $\mathbb{K}$, $\mathcal{FV}(\Omega)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $\Omega$ and $\mathcal{FV}(\Omega,E)$ be an $E$-valued counterpart of $\mathcal{FV}(\Omega)$
(where the term $E$-valued counterpart needs clarification itself).
For which spaces is it possible to lift series representations of elements of $\mathcal{FV}(\Omega)$ to elements of $\mathcal{FV}(\Omega,E)$?
We derive sufficient conditions for the answer to be affirmative which are applicable for many classical spaces of functions
$\mathcal{FV}(\Omega)$ having a Schauder basis.

BA-Vortrag