# 2007

**19 December 2007****Jorge Cortés**, Department of Mechanical and Aerospace Engineering, University of California, San Diego**Models, algorithms, and tools for distributed motion coordination**[slides pdf 5.6 MB, 49 pp]- Facultat de Matemàtiques i Estadística, aula 007; 15:30 h
**Abstract:**Cooperative robotic networks present new challenges that lie at the confluence of communication, computing, sensing, and control. A lot is known about the individual components of these networked systems, and yet novel theoretical developments are needed to integrate these components into autonomous networks with predictable behavior.

The objective of this talk is to present recently developed modeling, analysis, and design tools for motion coordination of cooperative networks. In our exposition, we pay special attention to the characterization of the correctness, performance, and cost of coordination algorithms. We illustrate our technical approach in disk-covering and sphere-packing deployment problems as well as in aggregation and consensus scenarios.

**30 November 2007**Ph.D./master course**David Martín de Diego**, Consejo Superior de Investigaciones Científicas, Madrid**Optimal control on Lie groups**(II)- Facultat de Matemàtiques i Estadística, aula 007; 12-14 h
**Contents**After reviewing the main part of the course, given in May, the discrete case will be studied.- Discrete variational mechanics. Construction of integrators preserving symplecticity and momentum.
- Discrete optimal control on Lie groups.

**21 November 2007****Javier de Lucas**, Department of Theoretical Physics, University of Zaragoza**Fundamentals and applications of Lie systems**- Facultat de Matemàtiques i Estadística, aula 101; 12:15 h
**Abstract:**We review the theory of Lie systems [LS,Wi] in the geometric point of view developed in [CGMa,CGM] in order to explain some of the applications of this formalism in integrability conditions of differential equations [CGR,CLRb], or in some physical applications [CLRa,CR].**References**- [LS] Lie S., "Vorlesungen uber continuierliche Gruppen mit Geometrischen und anderen Anwendungen", edited and revised by G. Scheffers, Teubner, Leipzig, 1893.
- [Wi] P. Winternitz, "Lie groups and solutions of nonlinear differential equations", in Nonlinear Phenomena, K.B. Wolf ed., LNP 189, Springer-Verlag, New York, 1983.
- [CGMa] J. F. Cariñena, J. Grabowski and G. Marmo, "Superposition rules, Lie Theorem and partial differential equations"; ArXiv: math-ph 0610013 (2006).
- [CGM] J.F. Cariñena, J. Grabowski and G. Marmo,
*Lie-Scheffers systems: a geometric approach*, Bibliopolis, Napoli, 2000. - [CGR] J.F. Cariñena, J. Grabowski and A. Ramos, "Reduction of time-dependent systems admitting a superposition principle", Acta Appl. Math. 66 (2001) 67-87.
- [CLRb] J.F. Cariñnena, J. de Lucas and A. Ramos, "A geometric approach to integrability conditions for Riccati equations", Electron. J. Diff. Equations 122 (2007) 1-14.
- [CLRa] J.F. Cariñnena, J. de Lucas and M.F. Rañada, "Nonlinear superposition rules", in Differential Geometric methods in Mechanics and Field Theory, eds. Cantrijn, M. Crampin, and B. Langerock, Academia Press (2007).
- [CR] J.F. Cariñnena and A. Ramos, "A new geometric approach to Lie systems and physical applications", Acta Appl. Math. 70 (2002) 43-69.

**20-29 November 2007**master course**José F. Cariñena**, Department of Theoretical Physics, University of Zaragoza**The geometry of classical and quantum mechanics: symplectic geometry**- Facultat de Matemàtiques i Estadística, aula 103.

Schedule:

Tuesday 20 16-18, Wednesday 21 16:30-18:30, Thursday 22 16-18, Friday 23 12-14;

Monday 26 16:30-18:30, Tuesday 27 16-18, Wednesday 28 16:30-18:30, Thursday 29 16-18. **Contents**The techniques specific to symplectic mechanics and Poisson structures that were developed in the second half of the past century have provided traditional classical mechanics with a strong and solid framework, and have allowed a better and deeper understanding of the problems, with a clear distinction between local and global characteristics of the ingredients of the theory. Reduction techniques have also proved to be very useful in the resolution of specific problems.

After nearly a century of existence, it seems appropriate to analyse quantum mechanics from a similar geometric perspective. The phase space of the classical system is replaced by a Hilbert space, which is endowed in a natural way with a symplectic structure, so that Schrödinger equation is nothing but the corresponding Hamilton equation. Reduction techniques of such systems also play an important role, mainly when one takes into account the character of rays, rather than vectors, of pure quantum states. This geometric formulation is paramount throughout the theory of quantum information and quantum control, whose present significance is beyond doubt.- A brief overview on some issues of classical mechanics
- Some important tools of differential geometry
- Hamiltonian dynamical systems
- Groups of symplectomorphisms of manifolds
- Some results on the theory of reduction
- Natural geometric structures of Hilbert spaces
- Geometric approach to quantum mechanics
- Mixed states and density operator
- A simple example: two-level systems

**7 November 2007****Narciso Román Roy**, Department of Applied Mathematics IV, UPC, Barcelona**An introduction to symmetries of**[notes of both talks pdf 220 kB, 15 pp]*k*-symplectic field theories- Campus Nord UPC, edifici C3, aula 005; 12 h
**Abstract:**This talk is devoted to giving some basic ideas about conservation laws and symmetries for first-order classical field theories, in the framework of the*k*-symplectic formulation. Different kinds of symmetries will be introduced, especially those leading to a geometrical statement of Noether's theorem, both in Hamiltonian and Lagrangian formalisms.

**31 October 2007****Narciso Román Roy**, Department of Applied Mathematics IV, UPC, Barcelona*k*-symplectic formulation of field theories- Campus Nord UPC, edifici C3, aula 005; 12 h
**Abstract:**There are several alternative models allowing a geometric description of (first order) classical field theories: polysymplectic,*k*-symplectic,*k*-cosymplectic, multisymplectic... In this talk we give a presentation of the most conceptually simple of them: the so-called*k*-symplectic formulation of field theory. With it, a geometric framework can be devised to describe lagrangian and hamiltonian formalisms for a certain type of theories. This model is a direct generalisation of the symplectic formalism used to describe geometrically autonomous mechanical systems.- Introduction.
- Geometric elements:
*k*-tangent and*k*-cotangent bundles of a manifold. - Lagrangian formalism.
- Hamiltonian formalism.

**17 October 2007****María Barbero Liñán**, Department of Applied Mathematics IV, UPC, Barcelona**Looking for strict abnormality in optimal control problems**- Campus Nord UPC, edifici C3, aula 005; 12 h
**Abstract:**Dubins car [1] is an example of control system. Apart from studying the trajectories of the system, we can also look for those trajectories that minimize a functional defined by the integral of a cost function. Therefore we get in optimal control theory [1,2,5]. A usual technique to solve optimal control problems is to use Pontryagin's maximum principle [1,2,5,6], stated in the classic and presymplectic versions [3,4,8] in this talk. In general, the maximum principle only provides with candidates to be optimal, called extremal. Among all the different kinds of extremals, the interest in the strict abnormal trajectories has increased recently because of the existence of strict abnormal minimizers [7,9] in subriemannian geometry [10].**References**- A. Agrachev and Y. Sachkov:
*Control Theory from the Geometric Viewpoint*, Springer-Verlag, Berlin-Heidelberg 2004. - F. Bullo and A. D. Lewis: Supplementary chapters for
*Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control*, Texts in Applied Mathematics 49, Springer-Verlag, New York, 2004. web page - M. Delgado-Tellez, A. Ibort, "A panorama of geometrical optimal control theory", Extracta Math. 18 (2003) 129-151.
- A. Echeverria-Enriquez, J. Marin-Solano, M.C. Munoz-Lecanda, N. Roman-Roy, "Geometric reduction in optimal control theory with symmetries", Rep. Math. Phys. 52 (2003) 89-113.
- V. Jurdjevic:
*Geometric Control Theory*, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, New York, 1997. - A. D. Lewis, "The maximum principle of Pontryagin in control and in optimal control", course held in Dept. of Applied Mathematics IV, Technical University of Catalonia, 9-16 May 2006. notes
- W. Liu, H. J. Sussmann, "Shortest paths for sub-Riemannian metrics on rank-two distributions", Mem. Amer. Math. Soc. 564, Jan. 1996.
- E. Martinez, "Reduction in optimal control theory", Rep. Math. Phys. 53 (2004) 79-90.
- R. Montgomery, "Abnormal Minimizers", SIAM J. Control Optim. 32 (1994) 1605-1620.
- R. Montgomery, "A survey of singular curves in sub-Riemannian geometry", Journal of Dynamical and Control Systems 1 (1995) 49-90.

- A. Agrachev and Y. Sachkov:

**10 October 2007****Xavier Gràcia**, Department of Applied Mathematics IV, UPC, Barcelona**The Hamilton-Jacobi equation in geometric mechanics**- Campus Nord UPC, edifici C3, aula 005; 12 h
**Abstract:**Hamilton-Jacobi theory is a special topic in hamiltonian dynamics that has spread into other areas of physics and mathematics. In this talk we review how the Hamilton-Jacobi equation appears form the study of canonical transformations, we study its geometrical formulation, and its generalisation in several directions. In particular, we study the possibility of representing all the solutions of a second-order differential equation by means of a family of first-order differential equations. We also investigate the relationship between the solutions of the generalised Hamilton-Jacobi equation and the symplectic structure of phase space.**References**- R. Abraham and J.E. Marsden,
*Foundations of mechanics*, 1978. - P. Libermann and C.-M. Marle,
*Symplectic geometry and analytical mechanics*, 1987. - V.I. Arnol'd,
*Mathematical methods of classical mechanics*, 1989. - J.V. José and E. Saletan,
*Classical dynamics. A contemporary approach*, 1998. - J.F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M.C. Muñnoz-Lecanda and N. Román-Roy, "Geometric Hamilton-Jacobi theory",
*Int. J. Geometric Methods Mod. Phys.***3**(2006) 1417-1458.

- R. Abraham and J.E. Marsden,

**8-11 May 2007**Ph.D./master course**David Martín de Diego**, Consejo Superior de Investigaciones Científicas, Madrid**Optimal control on Lie groups**[lecture notes pdf 315 kB, 42 pp]- Facultat de Matemàtiques i Estadística, aula 101.

Tuesday and Thursday 12-14 h; Wednesday, Thursday and Friday 15:30-18 h **Contents**- Introduction.
- The group of rotations. The rigid body.
- The Euler-Poincaré equations. Rigid body dynamics.
- Optimal control. Optimal control on Lie groups.