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1999

10 November 1999
Eva Miranda, Department of Algebra and Geometry, University of Barcelona
Completely integrable systems and hamiltonian torus actions
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, 204a (biblioteca de matemàtiques); 15:30: Abstract: Let (M,w,H) be a regular completely integrable system and let X be its algebra of hamiltonian vector fields. It is known, due to Arnold-Liouville theorem, that in a neighbourhood of a compact leaf L of the foliation F associated to X, there exist action-angle variables. Moreover, these action-angle variables provide with a hamiltonian action of a torus and an affine structure in the space of leaves.
If the hamiltonian system is singular similar results can be obtained under reasonable assumptions.
In this talk, we will recall some theorems concerning hamiltonian torus actions (Atiyah-Guillemin-Stenberg convexity theorem, Delzant's theorem) and we will find a link between these results and the ones obtained in the singular framework. Furthermore, some results on the determination of the symplectic germ of (M,F,L) will be given.

6 October 1999
Igor Kanatchikov, Center of Theoretical Physics, Polish Academy of Sciences
De Donder-Weyl "pre-canonical" formalism: from geometry to quantization
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, 204a (biblioteca de matemàtiques); 18:00: Abstract: The talk is devoted to the description of the De Donder-Weyl (DW) theory in the calculus of variations as a manifestly covariant generalization of the hamiltonian formalism from mechanics to field theory. We introduce the notion of polysymplectic form, as a generalization of the symplectic form to DW framework, and present our construction of the Poisson bracket operation on differential forms. Then we outline the algebraic structures which are generated by this bracket and generalize the Poisson algebra in mechanics (those are generalizations of the Gerstenhaber algebra). We show how the above bracket allows to identify the canonically conjugate variables and to write DW canonical equations in Poisson bracket formulation. In conclusion we discuss a possible quantization scheme in field theory based on the quantization of the Poisson bracket of forms, explain the notion of "pre-canonical" formalism, and briefly outline a "pre-canonical" description of Nambu-Goto string using the Nambu triple bracket, also showing its relationship to the binary bracket of forms.

6 October 1999
Joaquim Gomis, Department of Structure and Constituents of Matter, University of Barcelona
Strings, branes and non-commutative gauge theories
Facultat de Matemàtiques i Estadística, Campus Sud UPC, edifici U, seminari 1 (ala dreta, planta baixa); 13:15: Abstract: String theory contains nonperturbative objects, like D-branes. Ordinary supersymmetric gauge theories can be formulated in terms of D-Branes. A relation between non-commuative geometry, open string theory and D-branes will also be discussed.

15 September 1999
Narciso Román, Department of Applied Mathematics and Telematics, UPC
Equations and symmetries in covariant hamiltonian field theories
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: We state the intrinsic form of the hamiltonian equations of first-order classical field theories, in terms of multivector fields. Using this formalism, we study several aspects of these equations, such as their integrability, and the existence and non-uniqueness of solutions. Furthermore, the existence of first integrals of these equations is analyzed, and the relation between Cartan-Noether symmetries and generalized symmetries of the system is discussed. Finally, Noether's theorem is stated, both the standard version and a generalization to include higher-order Cartan-Noether symmetries.

1 July 1999
Jaume Franch, Department of Applied Mathematics and Telematics, UPC
Linearization by prolongations
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: It will be presented different types of linearization for nonlinear control systems; namely: static feedback linearization, linearization by prolongations, and dynamic feedback linearization. The relationship among them will also be explained. More precisely, a necessary and sufficient condition for a system to be linearizable by prolongations will be given. This condition refers to the maximum number of integrators needed to linearize a control system. The bound presented improves the existent bounds in the literature. The procedure will be applied to an example that was thought to be not linearizable by prolongations until now.

20 May 1999
Enric Fossas, Department of Applied Mathematics and Telematics, UPC
Some control techniques. Examples on bilinear systems
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: Some of the simplest circuits studied in power electronics consist of a voltage source, an inductor, a capacitor, a load and a switch. The designer's aim is often to decide which law the switch must obey in order that the circuit provides at the output a prescribed voltage, as well as the values of the parameters R, L, C to be proposed.
These systems are modelled through a system of differential equations of the form dx/dt = (Ax+B)+(Cx+D)u; here u designs the switch, which in the actual system takes values in {0,1}.
The aim of the talk is to present several control techniques for these systems, inside their corresponding theoretical frameworks. In particular, the sliding mode control and some linearizing techniques will be introduced. Among the theoretical frameworks, the linear control systems will be considered as modules, and, in the general case, dynamical systems will be considered in the state space representation.

29 April 1999
Jesús Marín, Department of Economical Mathematics, University of Barcelona
Variational principles in mechanics: geometric aspects
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: A geometric description of variational principles in mechanics is given, with special attention to constrained systems. The equations of motion for the case of nonholonomic constraints are obtained in both the vakonomic and nonholonomic formalisms from a variational principle. Both formalisms are compared and it is proved that they coincide when the constraints are integrable. As an example of vakonomic mechanics, a formulation of the theory of optimal control is described.

15 April 1999 --internal seminar--
Javier Yániz, Department of Applied Mathematics and Telematics, UPC
Affine connections and constraints
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: The relationship between affine connections and distributions is studied. In particular, given an affine connection and a distribution, one can construct a family of affine connections which restrict to a vector bundle connection in the distribution. When the original connection is the Levi-Civita connection of a Riemannian metric one may derive some conservation laws.

11 March 1999
Eva Navarro, Institut de Robòtica i Informàtica Industrial, CSIC-UPC
On discrete and discretized nonlinear systems
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, biblioteca de matemàtiques, 204a; 15:30: Abstract: Nonlinear discrete-time systems are proposed to study, focusing our attention to concepts, mathematical tools and methologies normally used in control theory. For this purpose, we will need the discrete-time counterparts of the definitions we are used to dealing with in continuous time. In this line, two main problems are stated: 1. The effect of time sampling on certain system propierties, namely: relative degree, zero dynamics, feedback linearizability, validity of continuous control design techniques under sampling and so on. 2. Control problem solutions in discrete time, such as: static and dynamic feedback linearization, input-output and disturbance decoupling problems, observer design methods, robust control, Fliess' canonical forms, the passivity approach, etc.

11 February 1999
Narciso Román, Department of Applied Mathematics and Telematics, UPC
Lagrangian equations in field theory in terms of multivector fields and connections
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, aula 005; 15:30: Abstract: From the study of integrability of multivector fields on differential manifolds and the relationship between connections and multivector fields on jet bundles, a geometric formulation of lagrangian equations for (first order) classical field theory is established. This allows a qualitative study about integrability, existence, and non-uniqueness of solutions of such equations, and their features. (Though geometric language will be used --differential manifolds, jet bundles, connections, multivector fields-- coordinate expressions will be provided as often as possible in order to make the exposition more fluent.)

14 January 1999
Carles Batlle, Department of Applied Mathematics and Telematics, UPC
On the approximation of delay elements by feedback
Dep. de Matemàtica Aplicada i Telemàtica, Campus Nord UPC, edifici C3, aula 005; 15:30: Abstract: A procedure for obtaining proper rational approximants of the transfer function of a delayor is proposed, generalizing previous results. We pose a general feedback problem and obtain its general solution. Explicit computations of the generalized approximants are obtained in terms of Bernoulli numbers and it is found that they correspond to iterated resummations of the previously known approximants before truncation. The properties and frequency performance of the new rational approximants are studied and compared to those of the original ones.