A mixed formulation for the direct approximation of L2L^2-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara \& Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension

Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane

In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following qualitative form of Landis conjecture, for $W_1, W_2 \in L^{\infty}(\mathbb R^2;\mathbb R^2)$, $V \in L^{\infty}(\mathbb R^2;\mathbb R)$ and $u \in H_{\mathrm{loc}}^{1}(\mathbb R^2)$ a real-valued weak solution to $-\Delta u - \nabla \cdot ( W_1 u ) +W_2 \cdot \nabla u + V u = 0$ in $\mathbb R^2$, satisfying for $\delta>0$, $|u(x)| \leq \exp(- |x|^{1+\delta})$, $x \in \mathbb R^2$, then $u \equiv 0$. Our methodology of proof is inspired by the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov that have treated the equation $-\Delta u + V u = 0$ in $\mathbb R^2$. Nevertheless, several differences and additional difficulties appear. New weak quantitative maximum principles are established for the construction of a positive multiplier in a suitable perforated domain, depending on the nodal set of $u$. The resulted divergence elliptic equation is then transformed into a non-homogeneous $\partial_{\overline{z}}$ equation thanks to a generalization of Stoilow factorization theorem obtained by the theory of quasiconformal mappings, an approximate type Poincar\'e lemma and the use of the Cauchy transform. Finally, a suitable Carleman estimate applied to the operator $\partial_{\overline{z}}$ is the last ingredient of our proof.Comment: Comments welcom

Dynamic Bayesian Network for Time-Dependent Classification Problems in Robotics

This chapter discusses the use of dynamic Bayesian networks (DBNs) for time-dependent classification problems in mobile robotics, where Bayesian inference is used to infer the class, or category of interest, given the observed data and prior knowledge. Formulating the DBN as a time-dependent classification problem, and by making some assumptions, a general expression for a DBN is given in terms of classifier priors and likelihoods through the time steps. Since multi-class problems are addressed, and because of the number of time slices in the model, additive smoothing is used to prevent the values of priors from being close to zero. To demonstrate the effectiveness of DBN in time-dependent classification problems, some experimental results are reported regarding semantic place recognition and daily-activity classification

On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see~[A.V. Fursikov, O.Yu. Imanuvilov: {\it Controllability of Evolutions Equations,} Lectures Notes Series, Vol.~34, Seoul National University, 1996]; this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables $x_1$, $x_2$ and $t$ that is second--order in time and fourth--order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.Comment: 35 page

SOCIAL ORGANIZATION BASED ON CHAIN-NETWORK LOGIC TO PROMOTE THE EXPLORATION OF NATIVE AÇAÍ IN WESTERN BRAZILIAN AMAZON.

The present paper has the objective to expose a proposition of organization within a chain and network logic, aiming to potentiate the extraction of the Native Açaí Berry at the Western Brazilian Amazon rainforest. This exploratory study involves the municipalities of Porto Velho, Guajará-Mirim and Machadinho D’Oeste, at the Brazilian state of Rondônia, with primary data originating mostly from conservation areas at the lower Madeira River region. As a result, it was possible to infer that from the native Açai Berry, derives food, pharmaceuticals and cosmetics, for both local consumption and international markets. It was found that beyond Açai Berry plantations availability, the lower Madeira River provides better transport logistic, consumer market and greater possibility of interaction with middleman than most Açai production areas. As a conclusion, it is made a proposition of an organizational arrangement to strengthen the extrativist productive chain of the Native Açaí Berry, based on the network and chain logic, oriented towards an organization based upon social organizations, manufacturing regularization and marketing