Semilinear problems with right-hand sides singular at u = 0 which change sign

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u = 0u=0. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u = 0u=0, while no restriction on its growth at u = 0u=0 is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/ |u|1/∣u∣, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1/ |u|\right.^{\gamma }\right. with 0 < \gamma < 10<γ<1

Comportement asymptotique d’une poutre élastique fixée sur une petite partie de l’une de ses extremités

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter ε tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size εrε) of the second one; the Neumann boundary condition is imposed on the remainder of the boundary. We show that the result depends on rε, and that there are 3 critical sizes, namely rε=ε3, rε=ε, and rε=ε1/3, and in total 7 different regimes. We also prove a corrector result for each behavior of rε.Nous étudions le comportement asymptotique de la solution d'un problème d'élasticité linéaire anisotrope et hétérogène dans un cylindre dont le diamètre ε tend vers zéro. Le cylindre est fixé (condition de Dirichlet homogène) sur la totalité de l'une de ses extrémités, mais seulement sur une petite partie (de taille εrε) de l'autre base ; sur le reste de la frontière on a la condition de Neumann. Nous montrons que le résultat depend de rε, et qu'il existe 3 tailles critiques, à savoir rε=ε3, rε=ε et rε=ε1/3, et au total 7 comportements différents. Nous donnons un résultat de correcteur pour tous les comportements de rε.Dirección General de Investigació

Comportamiento asintótico de una viga elástica fijada en pequeñas zonas de uno de sus extremos

Estudiamos el comportamiento asintótico de una viga elástica delgada cuando su anchura, ε, tiende a cero. La viga está fijada en la totalidad de una de sus bases, mientras que en la otra, sólo lo está en la unión de N pequeñas zonas de talla εrε , r ε tendiendo a cero. Sobre el resto de la frontera se impone una condición de Neumann. El comportamiento depende de r ε , el número de zonas de fijación y su distribución. Para N = 1 aparecen tres tallas críticas, ε 3 , ε y ε 1/3 y por tanto siete regímenes distintos. Si r ε ¿ ε 3 el comportamiento es el mismo que cuando no existe la pequeña zona de sujeción. Si r ε À ε 1/3 el comportamiento es el que obtendríamos si fijáramos en toda la base. En los demás casos aparecen comportamientos intermedios. Para N ≥ 2 el resultado es diferente. Así, si las zonas se concentran alrededor de tres puntos no alineados sólo aparecen dos tallas críticas, ε 3 y ε. Esto prueba que es preferible fijar la viga alrededor de tres puntos no alineados de una base a hacerlo alrededor de tan sólo uno, aún cuando usemos una zona de mucho mayor grosor

State Representation Learning for Control: An Overview

Representation learning algorithms are designed to learn abstract features that characterize data. State representation learning (SRL) focuses on a particular kind of representation learning where learned features are in low dimension, evolve through time, and are influenced by actions of an agent. The representation is learned to capture the variation in the environment generated by the agent's actions; this kind of representation is particularly suitable for robotics and control scenarios. In particular, the low dimension characteristic of the representation helps to overcome the curse of dimensionality, provides easier interpretation and utilization by humans and can help improve performance and speed in policy learning algorithms such as reinforcement learning. This survey aims at covering the state-of-the-art on state representation learning in the most recent years. It reviews different SRL methods that involve interaction with the environment, their implementations and their applications in robotics control tasks (simulated or real). In particular, it highlights how generic learning objectives are differently exploited in the reviewed algorithms. Finally, it discusses evaluation methods to assess the representation learned and summarizes current and future lines of research

A new Savage-Hutter type model for submarine avalanches and generated tsunami

In this paper we present a new two-layer model of Savage-Hutter type to study submarine avalanches. A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e. g. water). The model is derived in a system of local coordinates following a non-erodible bottom and takes into account its curvature. We prove that the model verifies an entropy inequality, preserves water at rest for a sediment layer and their solutions can be seen as particular solutions of incompressible Euler equations under hydrostatic assumptions. Buoyancy effects and the centripetal acceleration of the grain movement due to the curvature of the bottom are considered in the definition of the Coulomb term. We propose a two-step Roe type solver to discretize the presented model. It exactly preserves water at rest and no movement of the sediment layer, when its angle is smaller than the angle of repose, and up to second order all stationary solutions. Finally, some numerical tests are performed by simulating submarine and sub-aerial avalanches as well as the generated tsunami

Finite elements approximation of second order linear elliptic equations in divergence formwith right-hand side in L1

In this paper we consider, in dimension d≥ 2, the standard P1P1 finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L1(Ω), we prove that the unique solution of the discrete problem converges in W1,q0(Ω)W01,q(Ω) (for every q with 1≤q 1.Ministerio de Ciencia y TecnologíaMarie Curie Intra-European Fellowship (6th European Community Framework Programme

Deducción y simulación numérica de un nuevo modelo de avalanchas submarinas

En este trabajo se estudia un nuevo modelo de tipo Savage-Hutter para avalanchas submarinas. El modelo obtenido es de tipo aguas someras bicapa en el que la capa superior representa al fluido y la capa inferior representa a un estrato de material granular. En la deducción del modelo se han tenido en cuenta diferentes leyes constitutivas en los tensores de esfuerzos, la porosidad del estrato de sedimento o roca y un término de fricción de tipo Coulomb. El modelo obtenido puede aplicarse en el estudio de problemas de avalanchas submarinas y algunos tipos de tsunamis. Finalmente se presentan la discretización del modelo obtenido mediante técnicas de vol´umenes finitos bien equilibrados y un ensayo numérico donde se modela una avalancha submarina

Nou enfocament per a l'anàlisi de la controvèrsia no resolta sobre l'energia nuclear

Un equip d'investigadors de l'ICTA-UAB ha desenvolupat un nou enfocament per estudiar les controvèrsies sobre la governança de les tecnologies, centrant-se en el cas de l'energia nuclear, una de les tecnologies més controvertides i, al mateix temps, més desplegades dels últims 60 anys. Segons els investigadors, la controvèrsia sobre l'energia nuclear pot ser entesa com una sèrie de desajustos entre les expectatives i l'experiència, resultat del desafiament de fer front als alts nivells d'incertesa inherents a aquesta tecnologia.Un equipo de investigadores del ICTA-UAB ha desarrollado un nuevo enfoque para estudiar las controversias sobre la gobernanza de las tecnologías, centrándose en el caso de la energía nuclear, una de las tecnologías más controvertidas y, al mismo tiempo, más desplegadas de los últimos 60 años. Según los investigadores, la controversia sobre la energía nuclear puede ser entendida como una serie de desajustes entre las expectativas y la experiencia, resultado del desafío de hacer frente a los altos niveles de incertidumbre inherentes a esta tecnología.A research team from ICTA-UAB has developed a new approach to study controversies over technology governance, applied to the case of nuclear power, which has been one of the most controversial and, at the same time, largely deployed man-made technology over the past 60 years. According to the researchers, the controversy over nuclear power can be understood as a series of mismatches between expectations and experience. These mismatches are the result of the challenge of dealing with the high levels of uncertainty inherent to this technology

Multilayer models for shallow two-phase debris flows with dilatancy effects

We present here a multilayer model for shallow grain-fluid mixtures with dilatancy effects. It can be seen as a generalization of the depth-averaged model presented in Bouchut et al. (2016) [6], that includes dilatancy effects by considering a two-layer model, a mixture grain-fluid layer and an upper fluid layer, to allow the exchange of fluid between them. In the present work the approximation of the mixture layer is improved including normal variations of the velocities and concentrations of the two phases thanks to the multilayer approach. In the model presented here dilatancy effects induce in particular a non-hydrostatic pressure for both phases related to the excess pore fluid pressure. Contrary to the single-layer model, the computation of this excess pore pressure entrains a serious difficulty due to the multilayer approach. We identified here one of the main numerical difficulty of solving two-phase shallow debris flows models: the strongly non-linear behaviour and abrupt changes of the excess pore fluid pressure when starting from non-equilibrium conditions. We propose a simplified approach to approximate the excess pore fluid pressure in the simple case of uniform flows in the downslope direction and quantify the error made. Our method makes it possible to introduce two or three layers in the normal directions with a reasonable approximation. Analytical solutions for uniform grain-fluid flows over inclined planes, with and without side wall friction, are calculated and compared to the proposed model. The presented model preserves the total solid granular mass as in [6]. In the numerical results, we observe that the proposed model with a two layer description of the mixture accurately represents the velocity measured at the surface of the mixture in the laboratory experiments. This is obviously poorly represented by the depth-averaged velocity in single-layer models while the other quantities (solid volume fraction, basal excess pore fluid pressure) are similar to those obtained with single-layer models. Our numerical results show a significant impact of the parameters involved in dilatancy law, in particular on the calculation of the time evolution of the excess pore fluid pressure