# 8th ICMAT International Summer School on Geometry, Mechanics and Control

### Introduction:

Welcome to VIII ICMAT International Summer School on Geometry, Mechanics and Control organised by the Geometry,  Mechanics and Control (GMC) Network.

The 8th edition of the International Summer School on Geometry, Mechanics and Control will be held in La Cristalera, Madrid, Spain, in June 30 - July 4, 2014.

The school is oriented to young researchers, Ph.D. and postdoctoral students in Mathematics, Physics and Engineering, in particular those interested in focusing their research on geometric control and its applications to mechanical and electrical systems, and optimal control. It is intended to present an up-to-date view of some fundamental issues in these topics and bring to the participants attention some open problems, in particular problems related to applications.  This year the courses will be delivered by:

This School is an activity within the ICMAT Severo Ochoa Excellence Program.

### Important dates:

• Application for scholarship no later than April 30, 2014.
• Abstract submission for short talks/posters no later than April 30, 2014.
• Registration no later than June 1, 2014.
• Accommodation no later than June 1, 2014.

### Courses:

• François Gay Balmaz, CNRS, Ecole Normale Supérieure - Paris, France: Applications of geometric mechanics: strands dynamics, constraints, and fluid-structure interaction.

Abstract: We will present several applications of geometric mechanics to continuum systems, by focusing on the case of geometrically exact strands. In the first part, we will describe the geometric formulation of strand dynamics, based on the Euler-Poincare reduction for symmetry breaking. Contrary to the usual Kirchhoff description, the exact geometric approach allows the inclusion of nonlocal interactions and the treatment of tree-like structures encountered in molecular dynamics. The second part will include the study of elastic rods with perfect rolling contact, via the use of the Lagrange-d'Alembert principle for nonholonomic mechanics. We will also present the associated nonholonomic wave equations and discrete models. A formulation of the reduction process in terms of Dirac structures will be given. As an example of a fluid-structure interaction problem, we will finally derive a three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The equations of motion are obtained by coupling the Euler-Poincare equations of the tube and the fluid, via a holonomic constraint. We discuss the linear stability of the system since this question has been the main focus of previous literature on the subject, namely, the well-known "garden hose instability". We will conclude the course with the study of explicit solutions of the system.

• Juan Pablo Ortega, CNRS, Université de Franche-Comté, France: Symmetric Hamiltonian Systems. Stability Methods and Applications.

Abstract: This course will be a self-contained presentation of the different techniques developed during the last two decades to study the stability of relative solutions in symmetric Hamiltonian systems. The first session will include a presentation of the mathematical technology available to describe the dynamics of these systems: momentum maps and Hamiltonian reduction, normal form theory, and the reconstruction equations. In the second session we will concentrate on energy-momentum based stability methods for relative solutions (equilibria and periodic orbits). We will conclude the course with the study of explicit applications to various systems that exhibit different degrees of dynamical complexity.       Slides 1.   Slides 2.

• Yuri B. Suris, Technische Universität Berlin, Germany: Pluri-Lagrangian systems: variational approach to integrable systems

Abstract: We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multidimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a $d$-form $L$ on an $m$-dimensional space (called multi-time, $m>d$), whose coefficients depend on a sought-after function $x$ of $m$ independent variables (called field), find those fields $x$ which deliver critical points to the action functionals $S_\Sigma =\int_\Sigma L$ for any $d$-dimensional manifold $\Sigma$ in the multi-time. We present a detailed theory of the case $d=1$, which corresponds to systems of commuting Hamiltonian flows in the case of continuous time and to systems of commuting symplectic maps in the case of discrete time. We demonstrate that closedness of the 1-form $L$ on solutions of a pluri-Lagrangian system is equivalent to involutivity of the system of common integrals (integrability). In the discrete time case $d=2$, we derive the main building blocks of the multi-time Euler–Lagrange equations, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. Then, we discuss the relation of the closedness of the 2-form $L$ on solutions with various notions of integrability. We conclude by a discussion of the (widely open) case of $d=3$.

### Schedule:

 Monday 30 Tuesday 1 Wednesday 2 Thursday 3 Friday 4 9:00-9:30 OPENING 9:30-11:00 GAY-BALMAZ SURIS GAY-BALMAZ SURIS ORTEGA 11:00-11:30 COFFEE COFFEE/POSTER COFFEE 11:30- 13:00 ORTEGA GAY-BALMAZ SURIS (11:30-12:30) ORTEGA SURIS 13:00-15:00 LUNCH EXCURSION LUNCH 15:00-16:00 GAY-BALMAZ SURIS GAY-BALMAZ 16:00-16:15 COFFEE 16:15-17:15 ORTEGA ORTEGA SPECIAL EVENT (17:00) 17:15-18:15 SHORT TALKS SHORT TALKS

## Monday, June 30, 2014

17:15-17:30 Hassan Najafi Alishah (Instituto Superior Técnico, Portugal): Hamiltonian Evolutionary Games.

17:30-17:45 Marta Kowalczyk (Nicolaus Copernicus University, Poland): Bifurcations of critical orbits of gradient equivariant equations.

17:45-18:00 Marta Farré (ICMAT, Spain): The inverse problem for Lagrangians on a Lie algebroid.

## Tuesday, July 1, 2014

17:15-17:30 Yuri Ashrafyan (Yerevan State University, Armenia): The inverse Sturm-Liouville problem with fixed boundary conditions.

17:30-17:45 Miguel Vaquero (ICMAT, Spain): Reduction of the Hamilton-Jacobi equation.

17:45-18:00 Natalya Kudryashova (Cambridge Center for Analysis, UK): Strategic optimisation in the Internet Search Market: Effect of users' dynamics.

### Registration:

Please fill in this form online before June 1, 2014.

Remember to send a CV to icmat_8gmcschoolicmat [dot] es if you are applying for a scholarship before April 30, 2014.

Remember to send a title and an abstract to icmat_8gmcschoolicmat [dot] es if you are interested in presenting a short talk or poster before April 30, 2014.

The standard registration fee for participants is 200 Euros.

The fee for students, postdocs and retired scientists is 100 Euros.

The registration fee includes: conference materials and coffee breaks. Payment should be made in cash at arrival.

### Lodging reservation:

The price of the rooms at Residencia La Cristalera is 76 euros per day and per person (full board).

The number of rooms that have been reserved by the organization is limited therefore we ask you to register and make the reservation as soon as possible (no later than June 1, 2014).

### Updated on July 4, 2014.

Name Affiliation Country
Abel Álvarez Álvarez University of Oviedo Spain
Alexis Arnaudon Imperial College London UK
Yuri Ashrafyan Yerevan State University Armenia
Lucía Búa Universidade de Santiago de Compostela Spain
Santiago Capriotti Universidad Nacional del Sur Argentina
Leonardo Colombo ICMAT Spain
Francisco Crespo Universidad de Murcia Spain
Viviana Alejandra Díaz Universidad Nacional del Sur Argentina
Marta Farré ICMAT Spain
Marine Fontaine University of Manchester UK
François Gay-Balmaz CNRS, Ecole Normale Supérieure - Paris France
Xavier Gràcia Technical University of Catalonia Spain
Lyudmila Grigorieva CNRS, Université de Franche-Comté France
Elisa Lavinia Guzmán Alonso ICMAT Spain
David Iglesias Ponte Universidad de La Laguna Spain
Fernando Jiménez Zentrum Mathematik, Technische Universität Germany
Victor Manuel Jiménez Morales Universidad de La Laguna Spain
Jorge Alberto Jover Galtier Universidad de Zaragoza Spain
Jair Koiller INMETRO Brazil
Marta Kowalczyk Nicolaus Copernicus University Poland
Natalya Kudryashova Cambridge Center for Analysis UK
Juan Margalef IEM-CSIC Spain
David Martín de Diego ICMAT Spain
Hassan Najafi Alishah Instituto Superior Tecnico Portugal
Clarice Netto Universidade Federal Fluminense Brazil
Juan Pablo Ortega CNRS, Université de Franche-Comté France
Pere Daniel Prieto Martínez Universitat Politècnica de Catalunya Spain
Xavier Ramos Olivé Technical University of Catalonia/University of Barcelona Spain
Xavier Rivas Universitat Politècnica de Catalunya Spain
Daniel Rodríguez Díaz Universidad de La Laguna Spain
Ana Rojo Universidad de Zaragoza Spain
Narciso Román Roy Technical University of Catalonia Spain
Yuri Suris Technische Universität Berlin Germany
Miguel Teixidó Román Technical University of Catalonia Spain
Mariusz Tobolski University of Warsaw Poland
Miguel Vaquero ICMAT Spain
Silvia Vilariño Fernández Centro Universitario de la Defensa Zaragoza Spain
Szymon Wrzesien University of Warsaw Poland

### Organizing Committee

• María Barbero Liñán (Universidad Carlos III de Madrid and Instituto de Ciencias Matemáticas, Spain)
• David Iglesias Ponte (Universidad de La Laguna, Spain)
• David Martín de Diego (Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Spain)

### Scientific Committee

• Anthony Bloch (University of Michigan, USA)
• Jair Koiller (Fundação Getulio Vargas, Brazil)
• Manuel de León (Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Spain)
• Juan Carlos Marrero (Universidad de La Laguna, Spain)
• Eduardo Martínez (Universidad de Zaragoza, Spain)
• Miguel Muñoz Lecanda (Universidad Politécnica de Cataluña, Spain)

### How to get here:

Arrival by plane

You can take a taxi from the airport, but this is an expensive choice (at least 90 Euros, but be sure to ask for the price beforehand). You can also take a taxi from the Airport to Plaza de Castilla (around 45 Euros) and then take a bus to Miraflores.

A good choice is taking the Metro from the Airport to Plaza de Castilla, and then a bus to Miraflores. At the airport, take the Metro (underground), Line 8 (pink) to "Colombia" (3 stations). Transfer in Colombia to line 9 (purple) to "Plaza de Castilla" (3 stations). This should take about 25 minutes. In any case you can ask for a free map of the Metro when you buy a ticket.

Once you have reached Plaza de Castilla there are a lot of bus stops between the two big inclined towers. Only one (number 725) goes to Miraflores de Sierra. Look for number 725 (Madrid - Plaza de Castilla to Miraflores de la Sierra, Bustarviejo). Here you can find the current timetable for bus 725.

A taxi from Plaza de Castilla to Miraflores costs minimum 70 Euros.

The trip to Miraflores de la Sierra takes approximately 60 minutes and departure is aprox. each hour from 7.30 a.m. to 23.15 p.m. Your stop is the next after "Soto del Real". Once you get off, you still have to walk around 20 minutes up along the main road to get to La Cristalera.

A one-way ticket to Miraflores by bus number 725 costs about 5.10 Euros.

Arrival by train

Coming from the North, the train station is usually Chamartin. From Chamartin, you can take the Metro (line 10: dark blue) to Plaza de Castilla but it is only one station and it is also easy to walk from Chamartin to Plaza Castilla. If you arrive at Atocha, you can take the Metro (line 1: light blue, 13 stations) to Plaza Castilla and follow the above mentioned instructions for the bus.

Arrival by car

Take the big periferial highway M-40. If you come from the south, Portugal, or the Zaragoza Highways, take direction North (N-I). If you come from the Burgos Highway (N-I) take the direction M-607 Tres Cantos- Colmenar Viejo. Go out of the M-40 to the M-607 Highway towards Tres Cantos-Colmenar viejo. When the Highway ends (roughly 30 km) take the direction Miraflores. After 30 km you will reach the village. Once there, follow the signs towards La Cristalera, or Puerto de la Morcuera. One km outside the village you will find the entrance to the residence on your left (but watch out for it since sign is not a big one and may be missed.

### Supported by:

 Ministerio de Ciencia e Innovación Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas Universidad de La Laguna Geometry, Mechanics and Control Network