Rodrigo Lecaros - Academia.edu (original) (raw)

Papers by Rodrigo Lecaros

HAL (Le Centre pour la Communication Scientifique Directe), May 31, 2016

In a recent paper, the authors investigated the controllability of an underwater vehicle immersed... more In a recent paper, the authors investigated the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid, assuming that the flow was irrotational. The aim of the present paper is to pursue this study by considering the more general case of a flow with vorticity. It is shown here that the local controllability of the position and the velocity of the underwater vehicle (a vector in R 12) holds in a flow with vorticity whenever it holds in a flow without vorticity.

arXiv (Cornell University), May 31, 2016

Ima Journal of Applied Mathematics, Jan 15, 2019

Siam Journal on Control and Optimization, 2017

The direct problem of water-wave equations is the problem of determining the surface and its velo... more The direct problem of water-wave equations is the problem of determining the surface and its velocity potential, in time T > 0, for a given initial profile and velocity potential, where the profile of the bottom, the bathymetry, is known. In this paper, we study the inverse problem of recovering the shape of the solid bottom boundary of an inviscid, irrotational, incompressible fluid from measurements of a portion of the free surface. In particular, given the water-wave height and its velocity potential on an open set, together with the first time derivative of the free surface, on a single time, we address the identifiability problem. Moreover we compute the derivatives with respect to the shape of the bottom, which allows us to obtain the optimality conditions for this inverse problem.

Mathematics of Computation, Aug 25, 2015

We analyze a model optimal control problem for a 2D scalar conservation law-the so-called inverse... more We analyze a model optimal control problem for a 2D scalar conservation law-the so-called inverse design problem-with the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.

European Physical Journal D, Nov 5, 2010

arXiv (Cornell University), Oct 30, 2020

Nonpharmaceutical interventions (NPI) such as banning public events or instituting lockdowns have... more Nonpharmaceutical interventions (NPI) such as banning public events or instituting lockdowns have been widely applied around the world to control the current COVID-19 pandemic. Typically, this type of intervention is imposed when an epidemiological indicator in a given population exceeds a certain threshold. Then, the nonpharmaceutical intervention is lifted when the levels of the indicator used have decreased sufficiently. What is the best indicator to use? In this paper, we propose a mathematical framework to try to answer this question. More specifically, the proposed framework permits to assess and compare different event-triggered controls based on epidemiological indicators. Our methodology consists of considering some outcomes that are consequences of the nonpharmaceutical interventions that a decision maker aims to make as low as possible. The peak demand for intensive care units (ICU) and the total number of days in lockdown are examples of such outcomes. If an epidemiological indicator is used to trigger the interventions, there is naturally a trade-off between the outcomes that can be seen as a curve parameterized by the trigger threshold to be used. The computation of these curves for a group of indicators then allows the selection of the best indicator the curve of which dominates the curves of the other indicators. This methodology is illustrated using indicators in the context of COVID-19 using deterministic compartmental models in discretetime, although the framework can be adapted for a larger class of models.

Differential Equations and Applications, 2012

arXiv (Cornell University), Sep 28, 2022

We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions... more We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is done by using a suitable Carleman estimate for such models where the discrete parameters h and ∆t are connected to the one of the large Carleman parameters.

Journal of Mathematical Physics, Aug 1, 2016

A hydraulic jump is a physical phenomenon commonly observed in nature such as in open channel flo... more A hydraulic jump is a physical phenomenon commonly observed in nature such as in open channel flows or spillways and is dependent upon the relation between the initial upstream fluid speed and a critical speed characterized by a dimensionless number F known as the Froude number. In this paper we prove the existence of hydraulic jumps for stationary water-waves as a consequence of the existence of bifurcation branches of non-flat liquid interfaces originated from each of a sequence of upstream velocities F 1 > F 2 > • • • > F r > • • • (F r → 0 as r → ∞). We further establish explicitly, for F > 0, F F r , r ∈ N, the existence and uniqueness of the solution of a perfect, incompressible, irrotational free surface flow over a flat bottom, under the influence of gravity; as well as the corresponding hydraulic jump. Published by AIP Publishing.

Springer eBooks, 2014

We analyze a model tracking problem for a 1D scalar conservation law. It consists in optimizing t... more We analyze a model tracking problem for a 1D scalar conservation law. It consists in optimizing the initial datum so to minimize a weighted distance to a given target during a given finite time horizon. Even if the optimal control problem under consideration is of classical nature, the presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We adapt the so-called alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. This method was previously applied successfully on the inverse design problem and that of identifying the nonlinearity in the equation. The efficiency of the method in comparison with more classical discrete methods is illustrated by several numerical experiments. Contents 1. Introduction 2 2. Existence of Minimizers 3 3. The Discrete Minimization Problem 4 4. Sensitivity analysis: the continuous approach 6 4.1. Sensitivity without shocks 6 4.2. Sensitivity of the state in the presence of shocks 8 4.3. Sensitivity of the cost in the presence of shocks 9 5. The alternating descent method 12 6. Numerical approximation of the descent direction 13 6.1. The discrete approach 14 6.2. The alternating descent method 15 7. Numerical experiments 16 7.1. Experiment 1 17 7.2. Experiment 2 17 8. Conclusions and perspectives 20 References 21

arXiv (Cornell University), Mar 7, 2013

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinit... more In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron's return method, we establish some local controllability results for the position and velocities of the underwater vehicle. Examples with six, four, or only three controls inputs are given for a vehicule with an ellipsoidal shape.

Mathematical Control and Related Fields, 2023

This paper studies the integral turnpike and turnpike in average for a class of random ordinary d... more This paper studies the integral turnpike and turnpike in average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon.

The IMA Journal of Applied Mathematics, 2018

Journal of Differential Equations

Inverse Problems

In this work, we are interested in analyzing the well-known Calderón problem, which is an inverse... more In this work, we are interested in analyzing the well-known Calderón problem, which is an inverse boundary value problem of determining a coefficient function of an elliptic partial differential equation from the knowledge of the associated Dirichlet-to-Neumann map on the boundary of a domain. We consider the discrete version of the Calderon inverse problem with partial boundary data; in particular, we establish logarithmic stability estimates for the discrete Calderón problem, in dimension d ⩾ 3 , for the discrete H −r -norm on the boundary under suitable a priori bounds. The proof of our main result is based on a new discrete Carleman estimate for the discrete Laplacian operator with boundary observations.

Mathematical Control and Related Fields

arXiv (Cornell University), Jul 11, 2016

In this paper we study the stability and deformation of structures, in particular the wall of an ... more In this paper we study the stability and deformation of structures, in particular the wall of an open pit mine is studied by using information obtained from a variety of remote sensors and some extra data, with a novelty approach considering the use of mathematical models and data mining techniques. In particular we present two models to help the study the slope stability of pit and the possible occurrence of movements. Primarily we present an static model for slow movements, which will help us identify areas of possible risks areas with time horizons of several months or years, depends on the available information, before the wall start moving, and secondly a dynamic short-term model, which help us to determine risks of collapse zones with several days in advance. We remark that this methodology can be a powerful tool to plain future actions in order to simulate possible scenarios considering the production plans.

ArXiv, 2020

For underground mine, the current usual technique for ore extraction is block caving, which gener... more For underground mine, the current usual technique for ore extraction is block caving, which generates and induces seismic activity in the mine. To understand block caving method is one of the most challenging problems in underground mining. This method relies on gravity to break and transport large amounts of ore and waste. The state of art in damage models is not able to represent the real effect of the mining in the rock mass since for example the damage appears in the bottom of the domain under consideration and with this is not possible recover the subsidence sees in the mine. In this paper we present the analysis and implementation of the shear-compression damage model applied to underground mining proposed in [4]. We propose a fast algorithm based in the usual alternated algorithm used in gradient damage models [16] and show that this new algorithm is faster than the usual algorithm. We show some numerical tests in 3D and present interested simulations for different damage law...