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- 2020 (19) (entfernen)
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- Englisch (19)
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- Extremwertstatistik (2)
- Nash-Gleichgewicht (2)
- Optimierung (2)
- Zetafunktion (2)
- optimal control (2)
- state constraints (2)
- A-priori-Wissen (1)
- ADMM (1)
- Algebraic signal processing (1)
- Algebraische Signalverarbeitung (1)
Institut
- Institut für Mathematik (19) (entfernen)
This cumulative dissertation is organized as follows:
After the introduction, the second chapter, based on “Asymptotic independence of bivariate order statistics” (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived.
Chapter 3 is from “Conditional tail independence in archimedean copula models” (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas.
“Generalized pareto copulas: A key to multivariate extremes” (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017.
The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing.
In this dissertation, we develop and analyze novel optimizing feedback laws for control-affine systems with real-valued state-dependent output (or objective) functions. Given a control-affine system, our goal is to derive an output-feedback law that asymptotically stabilizes the closed-loop system around states at which the output function attains a minimum value. The control strategy has to be designed in such a way that an implementation only requires real-time measurements of the output value. Additional information, like the current system state or the gradient vector of the output function, is not assumed to be known. A method that meets all these criteria is called an extremum seeking control law. We follow a recently established approach to extremum seeking control, which is based on approximations of Lie brackets. For this purpose, the measured output is modulated by suitable highly oscillatory signals and is then fed back into the system. Averaging techniques for control-affine systems with highly oscillatory inputs reveal that the closed-loop system is driven, at least approximately, into the directions of certain Lie brackets. A suitable design of the control law ensures that these Lie brackets point into descent directions of the output function. Under suitable assumptions, this method leads to the effect that minima of the output function are practically uniformly asymptotically stable for the closed-loop system. The present document extends and improves this approach in various ways.
One of the novelties is a control strategy that does not only lead to practical asymptotic stability, but in fact to asymptotic and even exponential stability. In this context, we focus on the application of distance-based formation control in autonomous multi-agent system in which only distance measurements are available. This means that the target formations as well as the sensed variables are determined by distances. We propose a fully distributed control law, which only involves distance measurements for each individual agent to stabilize a desired formation shape, while a storage of measured data is not required. The approach is applicable to point agents in the Euclidean space of arbitrary (but finite) dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic (and even exponential) stability. A similar statement is also derived for nonholonomic unicycle agents with all-to-all communication. We also show how the findings can be used to solve extremum seeking control problems.
Another contribution is an extremum seeking control law with an adaptive dither signal. We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. A key novelty of the approach is an adaptive choice of the frequency parameter. In this way, the task of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of a minimum, but leads to exact convergence. For the case of an output function with a global minimum and no other critical point, we prove global convergence.
Finally, we present an extremum seeking control law for a class of nonholonomic systems. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. This requires a two-fold approximation of Lie brackets on different time scales. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent space. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.
For an arbitrary complex number a≠0 we consider the distribution of values of the Riemann zeta-function ζ at the a-points of the function Δ which appears in the functional equation ζ(s)=Δ(s)ζ(1−s). These a-points δa are clustered around the critical line 1/2+i\(\mathbb {R}\) which happens to be a Julia line for the essential singularity of ζ at infinity. We observe a remarkable average behaviour for the sequence of values ζ(δ\(_a\)).
The starting point of the thesis is the {\it universality} property of the Riemann Zeta-function $\zeta(s)$
which was proved by Voronin in 1975:
{\it Given a positive number $\varepsilon>0$ and an analytic non-vanishing function $f$ defined on a compact subset $\mathcal{K}$ of the strip $\left\{s\in\mathbb{C}:1/2 < \Re s< 1\right\}$ with connected complement, there exists a real number $\tau$ such that
\begin{align}\label{continuous}
\max\limits_{s\in \mathcal{K}}|\zeta(s+i\tau)-f(s)|<\varepsilon.
\end{align}
}
In 1980, Reich proved a discrete analogue of Voronin’s theorem, also known as {\it discrete universality theorem} for $\zeta(s)$:
{\it If $\mathcal{K}$, $f$ and $\varepsilon$ are as before, then
\begin{align}\label{discretee}
\liminf\limits_{N\to\infty}\dfrac{1}{N}\sharp\left\{1\leq n\leq N:\max\limits_{s\in \mathcal{K}}|\zeta(s+i\Delta n)-f(s)|<\varepsilon\right\}>0,
\end{align}
where $\Delta$ is an arbitrary but fixed positive number.
}
We aim at developing a theory which can be applied to prove the majority of all so far existing discrete universality theorems in the case of Dirichlet $L$-functions $L(s,\chi)$ and Hurwitz zeta-functions $\zeta(s;\alpha)$,
where $\chi$ is a Dirichlet character and $\alpha\in(0,1]$, respectively.
Both of the aforementioned classes of functions are generalizations of $\zeta(s)$, since $\zeta(s)=L(s,\chi_0)=\zeta(s;1)$, where $\chi_0$ is the principal Dirichlet character mod 1.
Amongst others, we prove statement (2) where instead of $\zeta(s)$ we have $L(s,\chi)$ for some Dirichlet character $\chi$ or $\zeta(s;\alpha)$ for some transcendental or rational number $\alpha\in(0,1]$, and instead of $(\Delta n)_{n\in\mathbb{N}}$ we can have:
\begin{enumerate}
\item \textit{Beatty sequences,}
\item \textit{sequences of ordinates of $c$-points of zeta-functions from the Selberg class,}
\item \textit{sequences which are generated by polynomials.}
\end{enumerate}
In all the preceding cases, the notion of {\it uniformly distributed sequences} plays an important role and we draw attention to it wherever we can.
Moreover, for the case of polynomials, we employ more advanced techniques from Analytic Number Theory such as bounds of exponential sums and zero-density estimates for Dirichlet $L$-functions.
This will allow us to prove the existence of discrete second moments of $L(s,\chi)$ and $\zeta(s;\alpha)$ on the left of the vertical line $1+i\mathbb{R}$, with respect to polynomials.
In the case of the Hurwitz Zeta-function $\zeta(s;\alpha)$, where $\alpha$ is transcendental or rational but not equal to $1/2$ or 1, the target function $f$ in (1) or (2), where $\zeta(\cdot)$ is replaced by $\zeta(\cdot;\alpha)$, is also allowed to have zeros.
Until recently there was no result regarding the universality of $\zeta(s;\alpha)$ in the literature whenever $\alpha$ is an algebraic irrational.
In the second half of the thesis, we prove that a weak version of statement \eqref{continuous} for $\zeta(s;\alpha)$ holds for all but finitely many algebraic irrational $\alpha$ in $[A,1]$, where $A\in(0,1]$ is an arbitrary but fixed real number.
Lastly, we prove that the ordinary Dirichlet series
$\zeta(s;f)=\sum_{n\geq1}f(n)n^{-s}$ and $\zeta_\alpha(s)=\sum_{n\geq1}\lfloor P(\alpha n+\beta)\rfloor^{-s}$
are hypertranscendental, where $f:\mathbb{N}\to\mathbb{C}$ is a {\it Besicovitch almost periodic arithmetical function}, $\alpha,\beta>0$ are such that $\lfloor\alpha+\beta\rfloor>1$ and $P\in\mathbb{Z}[X]$ is such that $P(\mathbb{N})\subseteq\mathbb{N}$.
This dissertation investigates the application of multivariate Chebyshev polynomials in the algebraic signal processing theory for the development of FFT-like algorithms for discrete cosine transforms on weight lattices of compact Lie groups. After an introduction of the algebraic signal processing theory, a multivariate Gauss-Jacobi procedure for the development of orthogonal transforms is proven. Two theorems on fast algorithms in algebraic signal processing, one based on a decomposition property of certain polynomials and the other based on induced modules, are proven as multivariate generalizations of prior theorems. The definition of multivariate Chebyshev polynomials based on the theory of root systems is recalled. It is shown how to use these polynomials to define discrete cosine transforms on weight lattices of compact Lie groups. Furthermore it is shown how to develop FFT-like algorithms for these transforms. Then the theory of matrix-valued, multivariate Chebyshev polynomials is developed based on prior ideas. Under an existence assumption a formula for generating functions of these matrix-valued Chebyshev polynomials is deduced.
In this paper we consider the class (θA, B) of parameter-dependent linear systems given by matrices A ∈ ℂ\(^{nxn}\) and B ∈ ℂ\(^{nxm}\). This class is of interest for several applications and the frequently met task for such systems is to steer the origin toward a given target family f(θ) by using an input that is independent from the parameter. This paper provides a collection of necessary and sufficient conditions for ensemble reachability for these systems.
In the thesis at hand, several sequences of number theoretic interest will be studied in the context of uniform distribution modulo one. <br>
<br>
In the first part we deduce for positive and real \(z\not=1\) a discrepancy estimate for the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \),
where \(\gamma_a\) runs through the positive imaginary parts of the nontrivial \(a\)-points of the Riemann zeta-function. If the considered imaginary
parts are bounded by \(T\), the discrepancy of the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \) tends to zero like
\( (\log\log\log T)^{-1} \) as \(T\rightarrow \infty\). The proof is related to the proof of Hlawka, who determined a discrepancy estimate for the
sequence containing the positive imaginary parts of the nontrivial zeros of the Riemann zeta-function. <br>
<br>
The second part of this thesis is about a sequence whose asymptotic behaviour is motivated by the sequence of primes. If \( \alpha\not=0\) is real
and \(f\) is a function of logarithmic growth, we specify several conditions such that the sequence \( (\alpha f(q_n)) \) is uniformly distributed
modulo one. The corresponding discrepancy estimates will be stated. The sequence \( (q_n)\) of real numbers is strictly increasing and the conditions
on its counting function \( Q(x)=\#\lbrace q_n \leq x \rbrace \) are satisfied by primes and primes in arithmetic progessions. As an application we
obtain that the sequence \( \left( (\log q_n)^K\right)\) is uniformly distributed modulo one for arbitrary \(K>1\), if the \(q_n\) are primes or primes
in arithmetic progessions. The special case that \(q_n\) equals the \(\textit{n}\)th prime number \(p_n\) was studied by Too, Goto and Kano. <br>
<br>
In the last part of this thesis we study for irrational \(\alpha\) the sequence \( (\alpha p_n)\) of irrational multiples of primes in the context of
weighted uniform distribution modulo one. A result of Vinogradov concerning exponential sums states that this sequence is uniformly distributed modulo one.
An alternative proof due to Vaaler uses L-functions. We extend this approach in the context of the Selberg class with polynomial Euler product. By doing so, we obtain
two weighted versions of Vinogradov's result: The sequence \( (\alpha p_n)\) is \( (1+\chi_{D}(p_n))\log p_n\)-uniformly distributed modulo one, where
\( \chi_D\) denotes the Legendre-Kronecker character. In the proof we use the Dedekind zeta-function of the quadratic number field \( \Bbb Q (\sqrt{D})\).
As an application we obtain in case of \(D=-1\), that \( (\alpha p_n)\) is uniformly distributed modulo one, if the considered primes are congruent to
one modulo four. Assuming additional conditions on the functions from the Selberg class we prove that the sequence \( (\alpha p_n) \) is also
\( (\sum_{j=1}^{\nu_F}{\alpha_j(p_n)})\log p_n\)-uniformly distributed modulo one, where the weights are related to the Euler product of the function.
We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.
The work in this thesis contains three main topics. These are the passage from discrete to continuous models by means of $\Gamma$-convergence, random as well as periodic homogenization and fracture enabled by non-convex Lennard-Jones type interaction potentials. Each of them is discussed in the following.
We consider a discrete model given by a one-dimensional chain of particles with randomly distributed interaction potentials. Our interest lies in the continuum limit, which yields the effective behaviour of the system. This limit is achieved as the number of atoms tends to infinity, which corresponds to a vanishing distance between the particles. The starting point of our analysis is an energy functional in a discrete system; its continuum limit is obtained by variational $\Gamma$-convergence.
The $\Gamma$-convergence methods are combined with a homogenization process in the framework of ergodic theory, which allows to focus on heterogeneous systems. On the one hand, composite materials or materials with impurities are modelled by a stochastic or periodic distribution of particles or interaction potentials. On the other hand, systems of one species of particles can be considered as random in cases when the orientation of particles matters. Nanomaterials, like chains of atoms, molecules or polymers, are an application of the heterogeneous chains in experimental sciences.
A special interest is in fracture in such heterogeneous systems. We consider interaction potentials of Lennard-Jones type. The non-standard growth conditions and the convex-concave structure of the Lennard-Jones type interactions yield mathematical difficulties, but allow for fracture. The interaction potentials are long-range in the sense that their modulus decays slower than exponential. Further, we allow for interactions beyond nearest neighbours, which is also referred to as long-range.
The main mathematical issue is to bring together the Lennard-Jones type interactions with ergodic theorems in the limiting process as the number of particles tends to infinity. The blow up at zero of the potentials prevents from using standard extensions of the Akcoglu-Krengel subadditive ergodic theorem. We overcome this difficulty by an approximation of the interaction potentials which shows suitable Lipschitz and Hölder regularity. Beyond that, allowing for continuous probability distributions instead of only finitely many different potentials leads to a further challenge.
The limiting integral functional of the energy by means of $\Gamma$-convergence involves a homogenized energy density and allows for fracture, but without a fracture contribution in the energy. In order to refine this result, we rescale our model and consider its $\Gamma$-limit, which is of Griffith's type consisting of an elastic part and a jump contribution.
In a further approach we study fracture at the level of the discrete energies. With an appropriate definition of fracture in the discrete setting, we define a fracture threshold separating the region of elasticity from that of fracture and consider the pointwise convergence of this threshold. This limit turns out to coincide with the one obtained in the variational $\Gamma$-convergence approach.
A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
(2020)
In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.