## Control Theory – is it an exciting topic? Impressions from the Summer School on Control of Evolution Equations @ Wuppertal

From September 30 to October 4, 2018 the Summer School on Control of Evolution Equations took place in Wuppertal and I appreciated it very much to be one of its participants. The program was divided into two interesting topics. Both lecture sessions shared an open and very enjoyable atmosphere due to the enthusiasm of the lecturers, Hans Zwart (Twente) and Marius Tucsnak (Bordeaux). My participation in the summer school lead to this blog article about control theory.

Participants of the Summer School on Control of Evolution Equations @ Wuppertal

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 $$T$$.

A vibrating elastic string, fixed at its right end and free at its left. In our model, we influence our string via a force acting on the left end.

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.

## Wavys Werk und Davids Beitrag

#### Wer ist schon gern berechenbar? Ein Gespräch mit einer Welle über ihre Simulationen.

via GIPHY

Stimme aus dem Nichts: Warum hat Gott nicht gesprochen „Es werde Schall!“?

Ich: Keine Ahnung.

Stimme aus dem Nichts: Hat er schon, es hat nur niemand gehört.

Ich: Sehr witzig. Wer ist da und warum sehe ich dich nicht?

Stimme aus dem Nichts: Ich bin Wavy, die akustische Welle. Du kannst mich nicht sehen, sondern
nur hören.

Ich: Das ergibt Sinn. Ich heiße David. Es freut mich dich endlich kennen zu lernen. Ich habe meine
Doktorarbeit über dich geschrieben.

Wavy: Ach ja? Um was geht es darin?

Ich: Ich habe Fehlerabschätzungen für Finite Elemente Lösungen von Wellengleichungen mit
dynamischen Randbedingungen bewiesen.

Wavy: Wie bitte? Wenn du glaubst mich durch „Fachchinesisch“ beeindrucken zu können, hast du
dich geschnitten.

Ich: Natürlich nicht, verzeih mir. Dann lass mich mit einer Frage beginnen: Weißt du wie ein
Mikrofon funktioniert?

## What is Bifurcation Theory?

In the following I would like to present the main ideas of bifurcation theory along with some basic examples that illustrate the theory. For the sake of shortness let me formulate the fundamental question of bifurcation theory in an abstract way. Suppose you are given an equation $$F(x,\lambda)=0$$ with the property that $$x=0$$ is always a solution, the so-called trivial solution. Is it possible to find some $$\lambda^*$$ such that a sequence of nontrivial solutions converges to $$(0,\lambda^*)$$? In other words: Do nontrivial solutions bifurcate from the trivial solutions? In the following I will present three equations from analysis, linear algebra and ordinary differential equations showing that bifurcation theory is a topic worth studying! The reading requires some amount of advanced mathematics — do not hesitate to contact me if you need some additional explanations. By the way: Next semester I will give a lecture on that topic which is suited for master students or advanced bachelor students with a background in analysis, differential equations and possibly boundary value problems, see below for more information. Continue Reading →

## Divide and Conquer

The ‘Divide and Conquer’-principle was already used in the Roman foreign policy. The politicians and warlords of the ancient world noticed that a large group of people can be easier controlled or even conquered if you split them up into several smaller groups.

Quicksort algorithm: source
CC BY-SA3.0 Roland H.

Luckily the ancient times are long gone, but the principle remains, in particular in computer science.  Many algorithms are based on the idea of splitting a task into several easy-to-handle parts. After solving these simpler parts, the solution of the full problem needs to be reconstructed from the solutions of these subproblems. So, it is possible to ‘conquer’ a complex task.

A famous example is the well-known Quicksort algorithm to sort a list of numbers. You start by picking one number in the list (the pivot element) and compare it to every other number of the list. If the other number is smaller, we put it on the left side of our pivot element, if it is larger, we put it on the right side. Thus, we get two sublists, one with smaller and one with larger numbers than the pivot element. We continue to sort these sublists in the same way with new pivot numbers from the sublists. The complete list is sorted when all sublists are sorted.

You may wonder what this is all about and how this is related to our CRC?

## What’s a Waveguide?

Waveguides have been mentioned a few times in this blog (i. e. here and here), so it seems quite apt to explain (in laymans words) what a waveguide is. As any serious pundit, I will start by quoting the internets source of everlasting wisdom:

„A waveguide is a structure that guides waves, such as electromagnetic or sound waves.“

This, of course, does not really help, since the reader can guess this definition just from reading the word „waveguide“. But we learn, at least, that a waveguide is not a person but a structure, unlike – for example – a tour guide. Let’s try again to find out what a waveguide is by picking the word apart:

What’s a wave? Luckily, due to discovery of the wave-particle duality in the last century, the answer is quite clear: Almost everything is a wave. So, that’s done.