Perturbación y decaimiento en ecuaciones parabólicas no autónomas

Damos condiciones finas sobre el tamaño y la forma de una perturbación que consigue que un problema lineal no autónomo pase de ser neutralmente estable a exponencialmente estable. Estos resultados se aplican al estudio del comportamiento asintótico de ecuaciones logísticas no autónomas

Motion planning with dynamics awareness for long reach manipulation in aerial robotic systems with two arms

Human activities in maintenance of industrial plants pose elevated risks as well as significant costs due to the required shutdowns of the facility. An aerial robotic system with two arms for long reach manipulation in cluttered environments is presented to alleviate these constraints. The system consists of a multirotor with a long bar extension that incorporates a lightweight dual arm in the tip. This configuration allows aerial manipulation tasks even in hard-to-reach places. The objective of this work is the development of planning strategies to move the aerial robotic system with two arms for long reach manipulation in a safe and efficient way for both navigation and manipulation tasks. The motion planning problem is addressed considering jointly the aerial platform and the dual arm in order to achieve wider operating conditions. Since there exists a strong dynamical coupling between the multirotor and the dual arm, safety in obstacle avoidance will be assured by introducing dynamics awareness in the operation of the planner. On the other hand, the limited maneuverability of the system emphasizes the importance of energy and time efficiency in the generated trajectories. Accordingly, an adapted version of the optimal Rapidly-exploring Random Tree algorithm has been employed to guarantee their optimality. The resulting motion planning strategy has been evaluated through simulation in two realistic industrial scenarios, a riveting application and a chimney repairing task. To this end, the dynamics of the aerial robotic system with two arms for long reach manipulation has been properly modeled, and a distributed control scheme has been derived to complete the test bed. The satisfactory results of the simulations are presented as a first validation of the proposed approach.Unión Europea H2020-644271Ministerio de Ciencia, Innovación y Universidades DPI2014-59383-C2-1-

Optimal existence classes and nonlinear-like dynamics in the heat equation in Rd

We analyse the behaviour of solutions of the linear heat equation in R d for initial data in the classes M ε (Rd) of Radon measures with ∫ R d e − ε | x | 2 d | u 0 | 0 M ε (Rd) consists precisely of those initial data for which the a solution of the heat equation can be given for all time using the heat kernel representation formula. After considering properties of existence, uniqueness, and regularity for such initial data, which can grow rapidly at infinity, we go on to show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t →∞ (in fact we show that this behaviour is generic), and that one can prescribe the behaviour of u (0 ,t ) to be any real-analytic function γ ( t ) on [0 , ∞ )

Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods

Asymptotic behaviour for a phase field model in higher order Sobolev spaces

In this paper we analyze the long time behavior of a phase field model by showing the existence of global compact attractors in the strong norm of high order Sobolev spaces

Extremal equilibria for reaction-diffusion equations in bounded domains and applications

We show the existence of two special equilibria, the extremal ones, for a wide class of reaction–diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equationDGESDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

The sub-supertrajectory method. Application to the nonautonomous competition Lotka-Volterra model

In this paper we study in detail the pullback and forwards attractions to non-autonomous competition Lotka-Volterra system. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution for these models, which attracts any other complete bounded trajectory. For that we present the sub-supertrajectory tool as a generalization of the now classical subsupersolution method