arXiv subjects

Rodrigo Lecaros

Publications and source records attributed to Rodrigo Lecaros.

12 recordsLinked to original sources

arXiv2026

Controllability for semi-discrete semilinear stochastic parabolic operators

In \cite{LPP:2025}, it was shown that, in arbitrary dimension, the spatial semi-discretization of a controlled stochastic parabolic operator is generically not null-controllable. Nevertheless, $\phi$-null controllability results remain attainable. The present paper extends those results to semi-discrete semilinear stochastic operators in arbitrary dimension, whose nonlinearities may also depend on the first-order spatial derivatives. The approach relies on establishing a new Carleman estimate for the adjoint backward stochastic parabolic operator, which yields $\phi$-null controllability for the associated linear system via a duality argument. The semilinear case is handeld by means of a fixed-point argument. As particular cases, our results recover the one-dimensional linear results of \cite{zhao:2024}, the multidimensional linear results of \cite{LPP:2025}, and the semilinear one-dimensional framework of \cite{WZ:2025} in the absence of gradient dependence.

math.OC↗

arXiv2025

Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions

In this paper, we study two types of inverse problems for space semi-discrete stochastic parabolic equations in arbitrary dimensions. The first problem concerns a semi-discrete inverse source problem, which involves determining the random source term of the white noise in the semi-discrete stochastic parabolic equation using observation data of the solution at the terminal time and on an arbitrary open spatial subdomain over a time interval. The second problem addresses a semi-discrete Cauchy inverse problem, which involves determining the solution of the stochastic parabolic equation in a space-time subdomain, from measurements of the solution and the trace of its discrete spatial derivative on an arbitrary open subset of the lateral boundary over a time interval. The key tools are three new global Carleman estimates for the semi-discrete stochastic parabolic operator, one for interior observations and two for boundary observations (homogeneous and nonhomogeneous Dirichlet conditions). Applying these Carleman estimates, we obtain Lipschitz and H\"older stability for the first and second inverse problems, respectively.

math.AP↗

arXiv2025

Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators

This work addresses an inverse problem for a semi-discrete parabolic equation, consisting of identifying the right-hand side of the equation from solution measurements at an intermediate time and within a spatial subdomain. We apply this result to establish a stability estimate for a coefficient inverse problem involving the recovery of a spatially dependent potential function. Furthermore, we present a reconstruction algorithm for recovering this coefficient and provide a proof of its convergence. Our approach relies on a novel semi-discrete Carleman estimate in which the parameter is constrained by the mesh size. Due to the discrete terms arising in the Carleman inequality, this method naturally introduces an error term associated with the solution's initial condition.

math.AP↗

arXiv2025

Carleman estimate for semi-discrete stochastic parabolic operators in arbitrary dimension and applications to controllability

This paper considers a semi-discrete forward stochastic parabolic operator with homogeneous Dirichlet conditions in arbitrary dimensions. We show the lack of null controllability for a spatial semi-discretization of a null-controllable stochastic parabolic system from any initial datum. However, by proving a new Carleman estimate for its semi-discrete backward stochastic adjoint system, we achieve a relaxed observability inequality, which is applied to derivative $\phi$-null controllability by duality arguments.

math.OC↗

arXiv2024

Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements

We consider the Boussinesq-Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDEconstrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-{\alpha} method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.

math.NA↗

arXiv2022

Discrete Carleman estimates and application to controllability for a fully-discrete parabolic operator with dynamic boundary conditions

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 $\Delta t$ are connected to the one of the large Carleman parameters.

math.AP↗

arXiv2020

A shear-compression damage model for the simulation of underground mining by block caving

Block caving is an ore extraction technique used in the copper mines of Chile. It uses gravity to ease the breaking of rocks, and to facilitate the extraction from the mine of the resulting mixture of ore and waste. To simulate this extraction process numerically and better understand its impact on the mine environment, we study 3 variational models for damage, based on the gradient damage model of Pham and Marigo (2010). In these models, the damage criterion may exhibit an anisotropic dependence on the spherical and deviatoric parts of the stress tensor. We report simulations that satisfactorily represent the expected evolution of the stress field in a block caving operation.

math.OC↗

arXiv2020

Assessment of event-triggered policies of nonpharmaceutical interventions based on epidemiological indicators

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 discrete-time, although the framework can be adapted for a larger class of models.

math.OC↗

arXiv2020

An inverse problem for Moore-Gibson-Thompson equation arising in high intensity ultrasound

In this article we study the inverse problem of recovering a space-dependent coefficient of the Moore-Gibson-Thompson (MGT) equation, from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.

math.AP↗

arXiv2017

A numerical procedure and unified formulation for the adjoint approach in hyperbolic PDE-constrained optimal control problems

The present paper aims at providing a numerical strategy to deal with PDE-constrained optimization problems solved with the adjoint method. It is done through out a unified formulation of the constraint PDE and the adjoint model. The resulting model is a non-conservative hyperbolic system and thus a finite volume scheme is proposed to solve it. In this form, the scheme sets in a single frame both constraint PDE and adjoint model. The forward and backward evolutions are controlled by a single parameter $\eta$ and a stable time step is obtained only once at each optimization iteration. The methodology requires the complete eigenstructure of the system as well as the gradient of the cost functional. Numerical tests evidence the applicability of the present technique

math.OC↗

arXiv2016

Control of underwater vehicles in inviscid fluids--II. Flows with vorticity

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.

math.AP↗

arXiv2013

Control of Underwater Vehicles in Inviscid Fluids. I: Irrotational Flows

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.

math.AP↗