Cienciashttps://hdl.handle.net/11441/227742018-06-20T13:39:36Z2018-06-20T13:39:36ZControlabilidad de ecuaciones escalares parabólicas: análisis de diferentes técnicashttps://hdl.handle.net/11441/710802018-03-19T12:21:04Z2017-01-01T00:00:00ZControlabilidad de ecuaciones escalares parabólicas: análisis de diferentes técnicas
The controllability of parabolic partial differential equations has been a topic of increasing interest in the last decades that it has attracted to good part of the mathematical community dedicated to the control. This type of equations and systems arise, for example, in the study of diverse phenomena related to the Physics, the Biology or the Chemistry. In this work we try to analyze different technologies that have allowed to give positive results of controllability of parabolic equations. The first positive results of controllability of parabolic equations were obtained in 1971 (Arch. Rational Mech. Anal. 43, 1971) y 1974 (Quart. Appl. Math. 32, 1974/75) por H.O. Fattorini y D.L. Russe. These proved the boundary controllability of the unidimensional heat equation by means of the method of the moments. The general results of controllability
to zero of N-dimensional parabolic equations were obtained independently for G. Lebeau and L. Robbiano and for A. Fursikov in O. Imanuvilov in 1995 - 1996. The first work is valid for autonomous parabolic equations and the result obtains as consequence of a spectral inequality obtained as consequence of local inequalites Carleman’s. The second work is applicable to not autonomous parabolic equations and the positive result of controllability
is obtained from inequalites of Carleman for the attached operator. In this work we analyze the different technologies developed by the authors mentioned previously to prove the controllability of parabolic equations. The analysis will begin for the method of the moments, will happen for the study of global inequality of Carleman for uniformly parabolic operators and will finish with Lebeau-Robbiano’s method.
2017-01-01T00:00:00ZSemigrupos de operadores de composiciónhttps://hdl.handle.net/11441/710792018-03-19T12:12:23Z2017-09-01T00:00:00ZSemigrupos de operadores de composición
The main problem we consider in this work is the study of semigroups of composition operators on spaces of analytic functions on the unit disc. Mainly, we focus on the problem in Hardy spaces, Hp(D), and Bergman spaces, Ap
(D). We can divide this work in two different parts. Firstly, we study the concept of strongly continuous semigroup of operators on a Banach space, as well as the existence of its infinitesimal generator. Secondly, we pay attention to semigroups of composition operators on spaces of analytic functions on the unit disc and its construction from the concept of semigroup of analytic functions. If {ϕt}t≥0 is a semigroup of analytic self maps of the unit disc D with the composition as operation between them, and X is a Banach space of analytic functions on D, then Q(t)f = f ◦ ϕt, f ∈ X, t ≥ 0, defines a semigroup of composition operators whenever Q(t) ∈ B(X) for t ≥ 0. We will proof under what conditions they are strongly continuous and we will focus on cases
such as Hardy spaces and Bergman spaces. Finally, we will calculate its infinitesimal generator. Results of the first part are classic and they can be found in many traditional books of Functional Analysis, such as [Rudin, W. Functional analysis. McGraw-Hill, Inc. Second Edition, 1991], while the rest of this report collects current results, so we will study recently published works of different authors.
2017-09-01T00:00:00ZEstudio de convertidor ADC para aplicaciones de espectrometría de bioimpedanciashttps://hdl.handle.net/11441/694382018-01-24T08:59:40Z2017-01-01T00:00:00ZEstudio de convertidor ADC para aplicaciones de espectrometría de bioimpedancias
2017-01-01T00:00:00ZSuperfluid dynamics in the outer core of neutron starshttps://hdl.handle.net/11441/693812018-01-23T12:18:51Z2017-01-01T00:00:00ZSuperfluid dynamics in the outer core of neutron stars
The consensus between observations and theory is that neutron star shows a layered structure determined by an increasing density and temperature with depth. Quantum degeneracy of nuclear matter lays at the basis of the existence of these exotic objects, and the Bose condensation of neutron and proton pairs leads to the emergence of superﬂuidity in the star interior. However, the precise equation of state of nuclear matter under such extreme conditions, relating its pressure and density, are yet unknown, and many proposals compete to provide a plausible picture compatible with the observational constrains. Currently, the study of neutron stars is becoming an interdisciplinary subject which is receiving an increasing attention due to more available data from precise observations and the possibility of making realistic computational simulations of the superﬂuid dynamics. This latter feature is crucial to account for the observed low moment of inertia and for the cooling down of the neutron star. It also allows us to understand the extraordinarily regular rotation of pulsars or even their observed, sudden speed-ups (glitches). Hydrodynamical models including the coupling of a normal ﬂuid and a superﬂuid, or two coupled superﬂuids (neutronic superﬂuid and protonic superconductor) are being used to ﬁnd dynamical instabilities that could explain the astronomical observations. Our work belongs to the latter group and models the macroscopic structure of the outer core, with typical densities around the nuclear saturation density and temperatures of the order of 108 K. The coupling between the two overlapped condensates of fermionic pairs is due to the entrainment of neutron and protons, which results from the Galilean invariance of the whole system. An equation of state based on the short range interactions between nucleons (Skyrme type) is assumed in order for the model to be consistent with observations. The resulting nonlinear equations of motion are linearized and the modes, obtained in the long wavelength regime, which allows to neglect the proton-electron interaction, are analytically derived as a function of the nuclear matter density. Comparison between the analytical predictions and numerical simulations of the original nonlinar equations are made, and relations with astronomical observations are discussed.
2017-01-01T00:00:00Z