The International Summer School on Geometry, Mechanics and Control is oriented to Ph.D. students and t postdoctoral students with undergraduate studies in Mathematics, Physics or Engineering, in particular to those who want to begin its research in geometrical aspects of mechanics, numerical integration, field theory and control theory. In this sense, the courses could be a complement to Ph.D. programs of different Universities. The main pretension is to form and attract young researchers of contrasted quality in the International context in leader topics around Mechanics, Differential Geometry and Control.

The School is also open to professionals who want to attend advanced and specialized courses oriented to geometrical techniques in those fields. It  pretends to show an updated version of the knowledge of some basic problems in these topics and to present to the participants  open problems and, in particular,  the applications by means of specialized courses taught by the best international researchers in the respective fields.

The specific scientific profile of the School in 2007 is based on the research lines of Numerical Analysis (in particular, numerical integration), Geometric Mechanics and Control Theory with applications to Engineering, Robotics and Physics.

In 2007, the School will develop two research lines which will linked along the course.  In each of the lines, it will be proposed discussion sessions and presentation of open problems and research to be done in the future.

  • Numerical integration and Geometric Mechanics

During the last decade, a remarkable effort has been made in the construction of geometric integrators for Lagrangian and Hamiltonian systems. The main idea consists in looking for numerical methods which preserve one or more geometric invariants associated to those systems (energy, first integrals, symplecticity ...).


  • Optimal Control Theory: Applications to Engineering

The mathematical theory of control is a broad area which, from a mathematical point of view, deals with three fundamental problems in Control Theory: modelling, analysis and design. For that, it is used different areas of Mathematics, for instance: Differential and Symplectic Geometry, Stability Theory of Dynamical Systems, Complex Analysis, Differential Analysis, Functional Analysis, Complexity Theory, etc. As it is expected, this approach implies the appearance of new problems in all those areas which suppose a fruitful interaction between them.


[PDF file]


Minicourse: Geometric integration of Hamiltonian systems


 JM Sanz-Serna

 Departament of Applied Mathematics

 University of Valladolid



Contents: Numerical methods for the integration of ordinary differential equations have a long and distinguished history, but only flourished in connection with the digital computer fifty or sixty years ago. Typically those methods are universal in the sense that they may be applied to any differential system. While this feature has made it possible to build general-purpose software packages of wide applicability, it is clear that the one-size-fits-all approach cannot be optimal in all cases. It is therefore plausible to investigate whether special methods can be introduced to integrate restricted but significant special classes of differential systems. Often, the most salient feature of the special class under consideration is some geometric property of the solution flow and one attemps to design numerical methods that preserve that geometric property. This is the approach that originated in the eighties and now known as "Geometric integration". The most important example of geometric integration, both in terms of the  range of applications (statistical simulations of gases and liquids, macromulecules, quantum chemistry, celestial mechanics, etc.) and of the volume of existing literature is the case of Hamiltonian systems. These are characterized geometrically by the symplecticness of their flows and one tries to design numerical integrators in such a way that the numerical solution also provides a symplectic transformation in phase space. The underlying hope is that symplectic integrators will better mimic the dynamic properties of Hamiltonian systems, particularly in long term integrations.

In the minicourse I will first review the history of numerical methods (one hour) and then present some background on Hamiltonian systems (one hour). After these preliminaries I will present the families of available symplectic  integrators (three hours). The course will conclude by analyzing to what extent symplectic integrators outperform their classical counterparts.



Minicourse: Computational Geometric Mechanics and Control



 Melvin Leok

 Department of Mathematics

 Purdue University



  Material: File1 File2 and File3

Contents: Geometric mechanics is concerned with the use of differential  geometric and symmetry techniques in the study of Lagrangian and  Hamiltonian mechanics. This approach serves as the theoretical  underpinning of innovative control methodologies in geometric control  theory that allow the attitude of satellites to be controlled using  changes in its shape, as opposed to chemical propulsion. It is also  the basis for understanding the ability of a falling cat to always  land on its feet, even when released in an inverted orientation.

Such control algorithms rely critically on the geometric structure  inherent in the mechanical system, and it is therefore natural to develop numerical implementations that are based on discrete  analogues of Lagrangian and Hamiltonian mechanics, and discrete  differential geometric techniques. This provides a systematic  framework for constructing geometric structure-preserving integrators  and geometric controllers for mechanical systems.

The minicourse will commence with a motivational overview of discrete  geometry and mechanics, through the use of an illustrative example  (one hour), followed by a discussion of the Lagrangian formulation of  mechanics and its corresponding discretization (one hour). I will  then show how exterior calculus is a generalization of vector  calculus, introduce its discrete analogue, and discuss applications  (two hours). Matrix Lie groups will be introduced, and applied to the  construction of numerical schemes for rigid body dynamics (one hour),  and the associated optimal control problem will be discussed (one hour).

Prerequisites: This minicourse will build upon concepts introduced in the preceding  minicourse on "Geometric integration of Hamiltonian systems," but  will otherwise assume a familiarity with differential equations,  elementary mechanics, vector calculus, and matrix algebra.




Minicourse: Distributed motion coordination of robotic networks


 Jorge Cortés

 University of California

 Santa Cruz


 Talk 1 [PDF]; Talk 2 [PDF]; Talk 3 [PDF]; Talk 4 [PDF]; Talk 5 [PDF]


 Material: File

Contents: Motion coordination is a remarkable phenomenon in biological systems and an extremely useful tool in man-made groups of vehicles, mobile sensors and embedded robotic systems. Just like animals do, groups of mobile autonomous agents need the ability to deploy over a given region, assume a specified pattern, rendezvous at a common point, or jointly move in a synchronized manner. Robotic teams and large-scale swarms are being considered for a broad class of applications, ranging from environmental monitoring, to search and rescue operations, and space imaging. Biology provides clear evidence that large-scale groups of animals coordinate their motion in order to efficiently pursue a collective objective. The collective behavior arises from local interactions, driven by individual goals, and with limited information exchange. The emergence of complex global behavior from simple local rules is by itself fascinating, and has generated a large body of literature in biology, physics, mathematics, and computer science. Modern technological advances make the deployment of large groups of autonomous mobile agents with on-board computing and communication capabilities increasingly feasible and attractive. As a consequence, the interest of the control community for motion coordination has increased rapidly in the last few years. This course will present analysis and design tools for distributed motion coordination algorithms. The introduction of the mathematical analysis techniques and design methodologies presented in the course will be done through application setups and examples from cooperative control, mobile sensor networks, and multi-agent robotic systems. The broad objective of the course is to illustrate ways in which systems and control theory helps us analyze emergent behaviors in animal groups and design autonomous and reliable robotic networks. The course will begin with an introduction to distributed motion coordination algorithms in biology and engineering (1 hour). We will discuss some of the envisioned applications of robotic networks, and justify the need for modeling, analysis and design mathematical tools. Then, we will briefly discuss important notions from graph theory, distributed algorithms and linear iterations (1 hour). We will then be ready to model robotic networks and their interconnection topology, and introduce some complexity notions that characterize the execution of coordination algorithms (1 hour). The final lectures will be devoted to design and analyze cooperative strategies for different tasks, including rendezvous (1 hour), deployment (1 hour) and agreement (1 hour). In doing this, we will introduce beautiful mathematical tools that will help us analyze these problems.


Prerequisites: Familiarity with ordinary differential equations, dynamical systems and analysis.


References: S.Martínez, J. Cortés, and F. Bullo. Motion coordination with distributed information. IEEE Control Systems Magazine, 2006. Submitted. Available at˜jcortes





Talk: Optimal Control of PDEs



 Eduardo Casas

 University of Cantabria




Contents: This talk is an introduction to the Optimal Control Theory of Partial Differential Equations. We will formulate some different problems corresponding to distributed and boundary (Dirichlet and Neumann) controls of elliptic and parabolic equations. We will present the goals of the theory and the methods to achieve them will be exhibited through one example. Essentially we will consider the problem of the existence of a solution and its numerical approximation, providing error estimates for the discretization. The first and second order optimality conditions will be showed to be a key tool in the analysis of the control problem.





Monday, June 25

Tuesday, June 26

Wednesday June 27

Thursday, June 28

Friday, June 29


Registration (8:00-9:00) and Opening(9:00-9:30)






Geometric integration of Hamiltonian systems


Geometric integration of Hamiltonian systems

Geometric integration of Hamiltonian systems

Distributed motion coordination of robotic networks

Geometric integration of Hamiltonian systems


















Distributed motion coordination of robotic networks


Distributed motion coordination of robotic networks


Computational Geometric Mechanics and Control

Optimal Control of PDEs


Until  13:30


Computational Geometric Mechanics and Control











 13:00-13:30 Closing





Geometric integration of Hamiltonian systems

Distributed motion coordination of robotic networks



Computational Geometric Mechanics and Control


Computational Geometric Mechanics and Control












Distributed motion coordination of robotic networks




Computational Geometric Mechanics and Control


LAB: Computational Geometric Mechanics and Control




Contact: gmcnetatull [dot] es

To make the registration, please send the following information by e-mail to gmcnetatull [dot] es


First Name
Family Name
E-mail address

Are you a student or PhD student?

Do you want to present a Poster? 


         Title of the Poster



The registration fee

The standard registration fee for participants is 200 Euros
The fee for students and retired scientists is 100 Euros
The registration fee includes: conference materials, coffee breaks and lunches along the Workshop

Payment should be made by transference to the following bank account

Bank: Cajacanarias

Account: 2065 0067 67 3000223636

IBAN: ES72 2065 0067 6730 0022 3636


Please, ask your bank to write explicitly in your transference order the name "GMC + surname of the participant" and send confirmation (your name and transfer details) by e-mail to gmcnetatull [dot] es or by fax to the number +34922318145, no later than 31 May, 2007.

Financial Support:


A limited number of scholarships for PhD Student and advanced undergraduate students, that may cover registration fees, lodging expenses or/and travel expenses, will be provided by the organisers.

If you want to apply for a scholarship, please send us to here your CV (and grades certificate in case your are an undergraduate student) before April 30, 2007.



Lodging reservation:

Lodging reservation

The number of rooms which have been reserved by the organization is limited, therefore we ask you to make the reservation as soon as possible.

Hotel Las Rocas (Hotel****)



   Single room: 73,83 Euros (Room + Breakfast)  VAT included

   Double room: 44,94 Euros per person (Room + Breakfast) VAT included




Hotel Miramar (Hotel***)


 Single room: 50 Euros (Room + Breakfast)  VAT included

 Double room: 30 Euros per person (Room + Breakfast) VAT included



To make a hotel reservation, please send the following information by e-mail to gmcnetatull [dot] es


First name
Family name
E-mail address

 I want do the lodging reservation in the hotel...........  for ....... person(s)


Date of arrival      ........

Date of departure  ........



Family Name First Name Institution Country
Aragüés Muñoz Rosario University of Zaragoza Spain
Bajars Janis Centrum voor Wiskunde an Informatica Netherlands
Barbero  Maria Technical  University of Catalonia Spain
Bassi  Luca University of Bologna Italy
Bustillo Saiz Paula University of Cantabria Spain
Cortés Jorge Univ. of California (Santa Cruz) USA
de León Manuel CSIC, Madrid Spain
Díaz  Viviana Universidad Nacional del Sur Argentina
Dragulete Oana Mihaela EPFL Lausanne Switzerland
Ferraro  Sebastián CSIC, Madrid Spain
Grebow Daniel John Purdue University USA
Grillo Sergio Instituto Balseiro-Univ. Nacional de Cuyo Argentina
Hinic Ana Institute of Mathematics and Informatics Serbia
Iglesias  Ponte CSIC, Madrid Spain
Leok Melvin Purdue University USA
Long David North Carolina State University USA
Lugo-Villeda Luís Iván Sant'Anna School of Advanced Studies Italy
Lukasik Maciej Warsaw University Poland
Marrero Juan Carlos Univ. La Laguna Spain
Martín de Diego David CSIC, Madrid Spain
Martínez  Eduardo University of Zaragoza Spain
Martínez Campos Cédric CSIC, Madrid Spain
Moriano Cristina Universidad Antonio Nebrija Spain
Muñoz-Lecanda Miguel C. Technical University of Catalonia Spain
Padrón Edith University of La Laguna Spain
Quintana-Portilla Gema University of Cantabria Spain
Ramírez Jr.  Juan Purdue University USA
Roldán Charles Purdue University USA
Rosado Eugenia Universidad Politécnica de Madrid  Spain
Santa-María Megía Ignacio Universidad Complutense de Madrid  Spain
Santoso Jenny University of Stuttgart Germany 
Sanz-Serna J M University of Valladolid Spain
Shen Bo university Dortmund Germany 
Sosa Diana University of La Laguna Spain
Tejado Balsera Inés University of Extremadura Spain
Turhan Murat EPFL Lausanne Switzerland
Vankerschaver Joris University of Ghent Belgium
Zadeh Joseph Purdue University USA



Organizing Committee

Manuel de León Rodríguez,  (CSIC, Madrid)

David Martín de Diego, (CSIC, Madrid)

Edith Padrón Fernández, (University of La Laguna)


Organizers of the Scientific Programme

Juan Carlos Marrero (University of La Laguna)

David  Martín de Diego (CSIC, Madrid)

Eduardo Martínez (University of  Zaragoza)

Miguel Muñoz Lecanda (Technical University of Catalonia)






How to get here:

To arrive at Castro:

By car; use mappy or via Michelin to plan your trip.

If you arrive by plane, the best arrival airport is the one at Bilbao. From there, you must take a bus until TermiBus of Bilbao (frequence:  half an hour, price: 1,50 euros, duration: half an hour). You can also take a taxi (20 euros). Teletaxi: 944800909. To go from Bilbao to Castro-Urdiales you could take a bus of the company Bizkaibus-Encartaciones (tel: 94 636 34 24). There is a bus each 30 minutes and the price is 2.42 euros, which lasts around 50 minutes. You take this bus at TermiBus station. This bus makes several stops in Castro. You can ask the drive about the best stop to go to the Hotel Las Rocas and/or Miramar (or to the Beach Brazomar, see the street guide).


If you come to Castro through Bilbao, a taxi could be another option (around 50 euros; tel 944800909)

The bus company ALSA (tel: 902 42 22 42) have routes to Castro Urdiales from Oviedo, Santander, San Sebastián, Vitoria, Pamplona and Zaragoza.

 Where to go:

The conferences and coffee breaks will be at Centro Cultural "La Residencia", where CIEM is located.

Lodging will be at Hotel Las Rocas (in which the participants will have lunch)  and Hotel Miramar.

In this Castro Urdiales street guide, the hotels Las Rocas and Miramar are in the squares C8 and B8, respectively. The location of CIEM is at square B7 (marked with an "8").

Here you have a printable version, from squares B6 to C8.


More information: