The post Frequenzkämme appeared first on Wave phenomena.

]]>Janina hat ein Lehramtsstudium Mathematik/Physik am KIT absolviert. Als sie sich für ihre Zulassungsarbeit mit einem mathematischen Thema auseinandergesetzt hat, bekam sie Lust, die mathematische Seite ihrer Ausbildung zum Master Mathematik zu vervollständigen. Anschließend hat sie eine Promotionsstelle in der KIT-Fakultät für Mathematik angenommen, wo sie auch im Schülerlabor Mathematik tätig war.

Im Zentrum von Janinas Arbeit stehen Frequenzkämme, die analytisch und

numerisch untersucht wurden. Gerade an den numerisch bestimmten Fällen war die Arbeitsgruppe in der E-Technik besonders interessiert.

Frequenzkämme sind optische Signale, die aus vielen Frequenzen bestehen und mehrere Oktaven überspannen können. Sie entstehen beispielsweise indem monochromatisches Laserlicht in einen Ringresonator eingekoppelt wird und die resonanten Moden des Ringresonators angeregt werden. Durch Mischung und aufgrund des nichtlinearen Kerr-Effekts des Resonatormaterials werden Frequenzkämme mit unterschiedlichen Eigenschaften erzeugt. Die mathematische Beschreibung des elektrischen Feldes innerhalb des Ringresonators erfolgt durch die Lugiato-Lefever Gleichung.

Von besonderem Interesse sind dabei Solitonen-Kerrkämme, die aus im Resonator umlaufenden zeitlich und räumlich stark lokalisierten Solitonen-Impulsen entstehen. Solitonen-Kerrkämme zeichnen sich durch eine hohe Zahl an Kammlinien und damit eine große optische Bandbreite, durch geringes Phasenrauschen und durch eine hohe Robustheit aus.

Durch numerische Simulationen konnte Janina heuristische Schlussfolgerungen ziehen und die im Zeitbereich am stärksten lokalisierten Frequenzkämme bestimmen. Damit übertrug sie ihre analytischen Ergebnisse in den anwendungsorientierten

Bereich des Projektes.

Wer das Gespräch anhören möchte, findet es hier.

The post Frequenzkämme appeared first on Wave phenomena.

]]>The post Im Fokus: Fatima Goffi appeared first on Wave phenomena.

]]>

*How did you get into mathematics and what were the most significant steps in your career so far?*

After my baccalaureate I wanted to study Computer Science. The first year was common to Mathematics and Computer Science, but then I felt more attracted to studying Maths. I did the Bachelor’s and the Master’s degrees at the university of my home city Batna. After that I moved to Algiers to do my PhD. At the moment I am continuing my research as a postdoc at the KIT. Each step in my career was of big interest for me. The most significant one was when I decided to switch my future career from Computer Science to studying Mathematics.

*What do you like about being a postdoctoral researcher?*

At the end of each step in my career, I had always the feeling that the next one will be the first. This is why I think that being a postdoctoral researcher is an important experience for me to get more into research and to gain more experience.

*In the CRC wave phenomena are considered analytically as well as numerically. Which research project would you like to conduct within the CRC?*

I like this collaborative relation between the analytical and numerical aspects for studying wave phenomena within the CRC. In my research project, I am working mainly on the analytical side. The numerical part is covered by my collaborators.

*About 80 scientists are united within our CRC to deal with wave phenomena. To what extent do you hope to profit from that?*

Working in the CRC allowed me to meet and discuss with experts from analysis and numerics. I think it is the right place to specialize in my research field.

*Although Karlsruhe currently looks like a large construction site the location has a lot to offer. In addition to a newly renovated Math building and the Campus being directly in the city center you can enjoy your free time in the park or in one of the many restaurants and cafes. Did you already have time to settle in?*

The location of the Campus, especially the Math building, is the first thing that attracted me during my first days in this nice city. The daily live is easy, because the city center is only a few hundred meters away. It’s the same thing with the greenery surrounding the famous Karlsruher Schloss – which is also the view that I have from my office window. I am lucky to be here!

The post Im Fokus: Fatima Goffi appeared first on Wave phenomena.

]]>The post Happy New Year 2019 appeared first on Wave phenomena.

]]>The post Merry Christmas to all of you appeared first on Wave phenomena.

]]>*A wave equation solved on a Christmas tree domain.*

The post Merry Christmas to all of you appeared first on Wave phenomena.

]]>The post Control Theory – is it an exciting topic? Impressions from the Summer School on Control of Evolution Equations @ Wuppertal appeared first on Wave phenomena.

]]>After this short introduction, some of you might have noticed that I have not answered the question in the title – a summer school with great lectures does not necessarily mean that the topic itself is interesting or even exciting. Maybe you even want to add the following questions:

*What are the main issues and goals of control theory?*

*How is it related to the CRC on wave phenomena?*

As usual, the best way to motivate the concepts of control theory and try to answer all these questions is to use an example, which I encountered in the lectures of Marius Tucsnak and Hans Zwart. For the sake of simplicity and evidence, I do only select the single example of a one-dimensional vibrating string. Nevertheless, this summer school taught me that there are many other examples having interesting connections to various parts within our CRC. If you are further interested in the example, it can be found in chapters 6.2 and 11.2 of the book *Observation and Control for Operator Semigroups* by M. Tucsnak and G. Weiss (Birkhäuser 2009) as well as chapter 9.2 of the book *Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces* by B. Jacob and H. Zwart (Birkhäuser 2012). Note furthermore that control theory is a wide subject, so I will only motivate the three concepts of **observability**, **controllability** and **stabilizability**.

Consider a vibrating elastic string (of your favorite length, say 1) that is fixed at both ends, which models for instance the string of a violin. First, there are no external forces acting on it (no gravity, damping, etc.). We call the vibrating string our system and the amplitude of its oscillations behaves according to a one-dimensional wave equation. Imagine now, that we do not know anything about the state of our system, i.e., its initial and current shape is not visible. Consequently, our system has become a black box and we cannot deduce anything about it.

So, we assume that we can measure an output of our system, say the slope at its left end (i.e., the spatial derivative at this point). One natural question concerns (exact) **observability**: can we deduce the initial (and thus the current) state of our system by means of our observations? The answer is yes, but we must observe our system on time scales larger than 2 which is also called the minimal observation time.

However, observing is not sufficient for most applications – acting (or controlling) is better. Thus, we follow up the path of control theory and get to the aspect of **controllability**. The goal of (exact) controllability is to move our system to a given state within a given time. The port over which we can manipulate the system is called input. That definition sounds quite abstract, so how is controllability related to our string? Let us unloose our string at the left end, so that it can move freely here. We now manipulate our system by means of a force that acts on the loose left end of the string and is caused by a damper or an engine. In this case, our system is exactly controllable for every time larger than 2 and this means the following. Suppose, we have a given initial shape \(w_{\text{start}}\), an initial velocity \(\partial_t w_{\text{start}}\) and a certain desired final shape \(w_{\text{final}}\) as well as a final velocity \(\partial_t w_{\text{final}}\) that our system should attain at a time \(T\) larger than 2. Then we can find a time-dependent force that acts on the loose left end of our string during the time \([0,T]\) and drives its state from \((w_{\text{start}},\partial_t w_{\text{start}})\) to the desired one \((w_{\text{final}},\partial_t w_{\text{final}})\) at time \(T\).

Our system with a damping force acting on the left end of our string is also a suitable instance of the concept of **stabilization**. Can we design a damper in such a way that the oscillations of our string decay in a uniform way, say exponentially, on large time intervals? Let us assume that we can observe also the vertical velocity of our string at the left end. Now we construct a damper that induces a force being negatively proportional to the string’s velocity at its left end. In this way we have designed a closed loop system, i.e., we have constructed a control (our input force) based on the knowledge of a given output (the observable velocity at the left end). Returning to our goal of stabilization: the oscillations of our closed loop system decay indeed in a uniform exponential way. Of course, there are many other ways of designing controls based on certain outputs and one can also ask for cost-optimal controls and the numerical approximation of controls.

During summer school, we considered many other examples of control theory for wave equations such as coupled strings, vibrating plates and other higher dimensional wave equations on various domains. Most fascinating for me is the interplay of abstract functional analysis with complex analysis having in mind real applications such as the design of machines by engineers – think of the manufacture of integrated circuits for computer chips requiring machines that are moving fast with extreme precision. In this way we have returned to the question from the title and I hope, I could convince you that control theory is indeed an exciting topic and worth to deal with.

The post Control Theory – is it an exciting topic? Impressions from the Summer School on Control of Evolution Equations @ Wuppertal appeared first on Wave phenomena.

]]>