Active control of a clamped beam equipped with piezoelectric actuator and sensor using generalized predictive control

of a flexible structure is here presented. The studied structure is a clamped-free beam equipped with collocated piezoelectric actuator/sensor. Piezoelectric transducers advantages lie in theirs compactness and reliability, making them commonly used in aeronautic applications, context in which our study fits. Theirs collocated placement allow the use of well-known control strategies with guaranteed stability. First an analytical model of this equipped beam is given, using the Hamilton's principle and the Rayleigh-Ritz method. After a review of the experimental setup (and notablv of the piezoelectric transducers), two control laws are described. The chosen one - Generalized Predictive Control (GPC) - will be compared to a typical control law in the domain of flexible structures, the Positive Position Feedback, one of the control lam mentioned above. Majors benefits of GPC lie in its robustness in front of model uncertainties and others disturbances. The results given come from experiments on the structure, performed thanks to a DSP. GPC appears to suit for the considered study's context (i.e. damping of the first vibration mode). Some improvements may, be reached. Among them, a more complex structure with more than a single mode to damp, and more uncertainties may be considered

New variables of separation for particular case of the Kowalevski top

We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.Comment: 11 pages, LaTeX with Ams font

On Charge-3 Cyclic Monopoles

We determine the spectral curve of charge 3 BPS su(2) monopoles with C_3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.Comment: 34 pages, 16 figures. Revision: Abstract added and a few small change

Leonard Euler: addition theorems and superintegrable systems

We consider the Euler approach to construction and to investigation of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stackel systems.Comment: The text of the talk at International Conference Geometry, Dynamics, Integrable Systems, September 2-7, 2008, Belgrade, Serbia, LaTeX, 18 page

Computing supersingular isogenies on Kummer surfaces

We apply Scholten\u27s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $GF(p^2)$ and corresponding abelian surfaces over $GF(p)$. Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography