arXiv subjects

Jon Asier Bárcena-Petisco

Publications and source records attributed to Jon Asier Bárcena-Petisco.

9 recordsLinked to original sources

arXiv2026

On the Homogenization of Norm and Time Optimal Controls for Parabolic Equations with Oscillating Coefficients

We study homogenization of norm-optimal and time-optimal controls for a one-dimensional heat equation with rapidly oscillating periodic diffusion coefficients. The controls act on a fixed subinterval and are subject to pointwise constraints in space and time. We first derive final-state observability and null-controllability estimates that are uniform with respect to the oscillation scale. Combining these estimates with periodic homogenization and an abstract theory of optimal controls for parabolic systems, we prove convergence of the norm-optimal values and controls to their homogenized counterparts. We then deduce convergence of the optimal times and time-optimal controls. The controls converge weakly-star in the space of essentially bounded functions and strongly in every Lebesgue space with finite exponent, after extension to a common time interval when necessary.

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arXiv2026

Tracking controllability on moving targets for parabolic equations

In this paper, we study the tracking controllability of a 1D parabolic-type equation. Notably, with controls acting on the boundary, we seek to approximately control the solution of the equation at specific points of the domain. We prove that acting on one boundary point, we control the solution on one target point, whereas acting on two boundary points, we can control the solution on up to two target points. In order to do so, when the target is fixed, we study the controllability by minimizing the corresponding problem with duality results. Afterwards, we study the controllability on moving points by applying a transformation that takes the problem to a fixed target. Lastly, we also solve some of these control problems numerically and compute approximations of the solutions and the desired targets, which validates our theoretical methodology.

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arXiv2026

A theoretical analysis on the resolution of parametric PDEs via Neural Networks designed with Strassen algorithm

We construct a family of Neural Networks that approximate matrix multiplication operator for any activation function such that there exists a Neural Network which can approximate the scalar multiplication function. In particular, we use the Strassen algorithm to bound the number of weights and layers needed for such Neural Networks. This allows us to define another Neural Network for approximating the inverse matrix operator. Finally, we discuss how it can be applied to numerically solve elliptic PDEs.

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arXiv2026

Averaged Controllability of Time-Fractional Schrödinger Equations with Random Quantum Diffusivity

This paper addresses the problem of averaged controllability for the time-fractional Schrodinger equation, where the quantum diffusivity parameter is a random variable with a general probability distribution. First, by exploiting the analyticity of the Mittag-Leffler function and Muntz's theorem, we show that the simultaneous null controllability of the system can occur only for a countable set of realizations of the random diffusivity. In particular, this implies the impossibility of simultaneous null controllability for absolutely continuous random diffusivity. Next, we prove the lack of exact averaged controllability for absolutely continuous random variables, irrespective of the control time. Furthermore, we introduce a new two-parameter fractional characteristic function, which allows us to construct a class of random variables satisfying null averaged controllability at any time from any arbitrary sensor set of positive Lebesgue measure. This is achieved using an open-loop control belonging to L^\infty and independent of the random parameter. In particular, we obtain the null controllability of the fractional biharmonic diffusion equation. Finally, we conclude with several remarks and open problems that merit future investigation.

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arXiv2025

Averaged Controllability of the Random Schrödinger Equation with Diffusivity Following Absolutely Continuous Distributions

This paper is devoted to the averaged controllability of the random Schrödinger equation, with diffusivity as a random variable drawn from a general probability distribution. First, we show that the solutions to these random Schrödinger equations are null averaged controllable with an open-loop control independent of randomness from any arbitrary subset of the domain with strictly positive measure and in any time. This is done for an interesting class of random variables, including certain stable distributions, specifically recovering the known result when the random diffusivity follows a normal or Cauchy distribution. Second, by the Riemann-Lebesgue lemma, we prove for any time the lack of averaged exact controllability in a $L^2$ setting for all absolutely continuous random variables. Notably, this implies that this property is not inherited from the exact controllability of the Schrödinger equation. Third, we show that simultaneous null controllability is not possible except for a finite number of scenarios. Finally, we perform numerical simulations that robustly validate the theoretical results.

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arXiv2025

Fibonacci-like partitions and their associated piecewise-defined permutations

In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard Fibonacci-like partitions of the first kind. That partition refines the well-known partition in two parts associated to the Wythoff sequences. In a similar way, we introduce another family of Fibonacci-like partitions of the natural numbers, with an initial segment removed, in a number of parts which also follows the Fibonacci sequence, the standard Fibonacci-like partitions of the second kind. We study some piecewise-defined permutations over Fibonacci-like partitions, which give both torsion-free and torsion elements in the symmetric group over the set of natural numbers. We give an application of Fibonacci-like partitions to the study of a sequence analyzed by A. Shapovalov and by B.J. Venkatachala. We give also a generalization of the mentioned sequence, involving techniques that are not related to Fibbonaci numbers nor Beatty sequences, obtaining an infinite family of sequences that we conjecture that produce an infinite orthogonal array in which the number of symbols, trials and factors is countable infinite, and having an automorphism group isomorphic to Z acting regularly on the symbol set.

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arXiv2024

On uniform null controllability of transport-diffusion equations with vanishing viscosity limit

This paper aims to address an interesting open problem, posed in the paper "Singular Optimal Control for a Transport-Diffusion Equation" of Sergio Guerrero and Gilles Lebeau in 2007. The problem involves studying the null controllability cost of a transport-diffusion equation with Neumann conditions, where the diffusivity coefficient is denoted by $\varepsilon>0$ and the velocity by $\mathfrak{B}(x,t)$. Our objective is twofold. First, we investigate the scenario where each velocity trajectory $\mathfrak{B}$ originating from $\overlineΩ$ enters the control region in a shorter time at a fixed entry time. By employing Agmon and dissipation inequalities, and Carleman estimate in the case $\mathfrak{B}(x,t)$ is the gradient of a time-dependent scalar field, we establish that the control cost remains bounded for sufficiently small $\varepsilon$ and large control time. Secondly, we explore the case where at least one trajectory fails to enter the control region and remains in $Ω$. In this scenario, we prove that the control cost explodes exponentially when the diffusivity approaches zero and the control time is sufficiently small for general velocity.

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arXiv2024

Tracking controllability for the heat equation

We study the tracking or sidewise controllability of the heat equation. More precisely, we seek for controls that, acting on part of the boundary of the domain where the heat process evolves, aim to assure that the normal trace or flux on the complementary set tracks a given trajectory. The dual equivalent observability problem is identified. It consists on estimating the boundary sources, localized on a given subset of the boundary, out of boundary measurements on the complementary subset. Classical unique continuation and smoothing properties of the heat equation allow us proving approximate tracking controllability properties and the smoothness of the class of trackable trajectories. We also develop a new transmutation method which allows to transfer known results on the sidewise controllability of the wave equation to the tracking controllability of the heat one. Using the flatness approach we also give explicit estimates on the cost of approximate tracking control. The analysis is complemented with a discussion of some possible variants of these results and a list of open problems.

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arXiv2022

Exact controllability to the trajectories of the one-phase Stefan problem

This paper deals with the exact controllability to the trajectories of the one--phase Stefan problem in one spatial dimension. This is a free-boundary problem that models solidification and melting processes. It is assumed that the physical domain is filled by a medium whose state is liquid on the left and solid, with constant temperature, on the right. In between we find a free-boundary (the interface that separates the liquid from the solid). In the liquid domain, a parabolic equation completed with initial and boundary conditions must be satisfied by the temperature. On the interface, an additional free-boundary requirement, called the {\it Stefan condition,} is imposed. We prove the local exact controllability to the (smooth) trajectories. To this purpose, we first reformulate the problem as the local null controllability of a coupled PDE-ODE system with distributed controls. Then, a new Carleman inequality for the adjoint of the linearized PDE-ODE system, coupled on the boundary through nonlocal in space and memory terms, is presented. This leads to the null controllability of an appropriate linear system. Finally, a local result is obtained via local inversion, by using {\it Liusternik-Graves' Theorem}. As a byproduct of our approach, we find that some parabolic equations which contains memory terms localized on the boundary are null-controllable.

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