Contractive probability metrics and asymptotic behavior of dissipative kinetic equations

The present notes are intended to present a detailed review of the existing results in dissipative kinetic theory which make use of the contraction properties of two main families of probability metrics: optimal mass transport and Fourier-based metrics. The first part of the notes is devoted to a self-consistent summary and presentation of the properties of both probability metrics, including new aspects on the relationships between them and other metrics of wide use in probability theory. These results are of independent interest with potential use in other contexts in Partial Differential Equations and Probability Theory. The second part of the notes makes a different presentation of the asymptotic behavior of Inelastic Maxwell Models than the one presented in the literature and it shows a new example of application: particle's bath heating. We show how starting from the contraction properties in probability metrics, one can deduce the existence, uniqueness and asymptotic stability in classical spaces. A global strategy with this aim is set up and applied in two dissipative models.

Robust decoupled visual servoing based on structured light

This paper focuses on the problem of realizing a plane-to-plane virtual link between a camera attached to the end-effector of a robot and a planar object. In order to do the system independent to the object surface appearance, a structured light emitter is linked to the camera so that 4 laser pointers are projected onto the object. In a previous paper we showed that such a system has good performance and nice characteristics like partial decoupling near the desired state and robustness against misalignment of the emitter and the camera (J. Pages et al., 2004). However, no analytical results concerning the global asymptotic stability of the system were obtained due to the high complexity of the visual features utilized. In this work we present a better set of visual features which improves the properties of the features in (J. Pages et al., 2004) and for which it is possible to prove the global asymptotic stability

A mixed H2/H∞-based semiactive control for vibration mitigation in flexible structures

In this paper, we address this problem through the design of a semiactive controller based on the mixed H2/H∞ control theory. The vibrations caused by the seismic motions are mitigated by a semiactive damper installed in the bottom of the structure. It is meant by semiactive damper, a device that absorbs but cannot inject energy into the system. Sufficient conditions for the design of a desired control are given in terms of linear matrix inequalities (LMIs). A controller that guarantees asymptotic stability and a mixed H2/H∞ performance is then developed. An algorithm is proposed to handle the semiactive nature of the actuator. The performance of the controller is experimentally evaluated in a real-time hybrid testing facility that consists of a physical specimen (a small-scale magnetorheological damper) and a numerical model (a large-scale three-story building)

Stability and asymptotic analysis of a fluid-particle interaction model

We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models.

Asymptotic flocking dynamics for the kinetic Cucker-Smale model

In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

On dictatorship, economic development and stability

This paper aims to account for varying economic performances and political stability under dictatorship. We argue that economic welfare and social order are the contemporary relevant factors of political regimes' stability. Societies with low natural level of social order tend to tolerate predatory behavior from dictators in exchange of a provision of civil peace. The fear of anarchy may explain why populations are locked in the worst dictatorships. In contrast, in societies enjoying a relative natural civil peace, dictatorship is less likely to be predatory because low economic welfare may destabilize it.

Paths to stability for matching markets with couples

We study two-sided matching markets with couples and show that for a natural preference domain for couples, the domain of weakly responsive preferences, stable outcomes can always be reached by means of decentralized decision making. Starting from an arbitrary matching, we construct a path of matchings obtained from `satisfying' blocking coalitions that yields a stable matching. Hence, we establish a generalization of Roth and Vande Vate's (1990) result on path convergence to stability for decentralized singles markets. Furthermore, we show that when stable matchings exist, but preferences are not weakly responsive, for some initial matchings there may not exist any path obtained from `satisfying' blocking coalitions that yields a stable matching.

Candidate stability and voting correspondences

We study the incentives of candidates to enter or to exit elections in order to strategically affect the outcome of a voting correspondence. We extend the results of Dutta, Jackson and Le Breton (2000), who only considered single-valued voting procedures by admitting that the outcomes of voting may consist of sets of candidates. We show that, if candidates form their preferences over sets according to Expected Utility Theory and Bayesian updating, every unanimous and non dictatorial voting correspondence violates candidate stability. When candidates are restricted to use even chance prior distributions, only dictatorial or bidictatorial rules are unanimous and candidate stable. We also analyze the implications of using other extension criteria to define candidate stability that open the door to positive results.

On the stability of cooperation structures

Qin [J. Eco. Th., 1996] recently showed that in a game of endogenous formation of cooperation structure, if the underlying TU-game is superadditive, then the full cooperation structure is stable. In this note, we characterize the class of games that ensure the stability of the full cooperation structure, and show that this class is much larger than that of superadditive TU-games.

Reduction, Linearization and stability of relative equilibria for mechanical systems on riemannian manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory "commute." As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory.

Stability and manipulation in representative democracies

This paper is devoted to the analysis of all constitutions equipped with electoral systems involving two step procedures. First, one candidate is elected in every jurisdiction by the electors in that jurisdiction, according to some aggregation procedure. Second, another aggregation procedure collects the names of the jurisdictional winners in order to designate the final winner. It appears that whenever individuals are allowed to change jurisdiction when casting their ballot, they are able to manipulate the result of the election except in very few cases. When imposing a paretian condition on every jurisdictions voting rule, it is shown that, in the case of any finite number of candidates, any two steps voting rule that is not manipulable by movement of the electors necessarily gives to every voter the power of overruling the unanimity on its own. A characterization of the set of these rules is next provided in the case of two candidates.

Stability of periodic solutions obtained via the averaging method for nonsmooth Lipschitz system

"Vegeu el resum a l'inici del document del fitxer adjunt."

On the stability of periodic orbits in Rn

We consider an autonomous differential system in Rn with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of n ¡ 1 codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.

On the asymptotic expansion of the colored Jones polynomial for torus knots

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern-Simons invariant and Reidemeister torsion.