arXiv subjects

Cristóbal Loyola

Publications and source records attributed to Cristóbal Loyola.

6 recordsLinked to original sources

arXiv2026

Carleman estimates for the Korteweg-de Vries equation with piecewise constant main coefficient

In this article, we investigate observability-related properties of the Korteweg-de Vries equation with a discontinuous main coefficient, coupled by suitable interface conditions. The main result is a novel two-parameter Carleman estimate for the linear equation with internal observation, assuming a monotonicity condition on the main coefficient. As a primary application, we establish the local exact controllability to the trajectories by employing a duality argument for the linear case and a local inversion theorem for the nonlinear equation. Secondly, we establish the Lipschitz-stability of the inverse problem of retrieving an unknown potential using the Bukhge{\uı}m-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.

math.AP↗

arXiv2026

iDocV2: Leveraging Self-Supervision and Open-Set Detection for Improving Pattern Spotting in Historical Documents

Considering the imminent massification of digital books, it has become critical to facilitate searching collections through graphical patterns. Current strategies for document retrieval and pattern spotting in historical documents still need to be improved. State-of-the-art strategies achieve an overall precision of $0.494$ for pattern spotting, where the precision for small non-square queries reaches 0.427. In addition, the processing time is excessive, requiring up to 7 seconds for searching in the DocExplore dataset due to a dense-based strategy used by SOTA models. Therefore, we propose a new model based on a better encoder (iDoc), trained under a self-supervised strategy, and an open-set detector to accelerate searching. Our model achieves competitive results with state-of-the-art pattern spotting and document retrieval, improving speed by 10x. Furthermore, our model reaches a new SOTA performance on the small non-square queries, achieving a new precision of 0.612.Different from the previous version, this leverages non-maximum suppression to reduce false positives.

cs.CV↗

arXiv2026

Global propagation of analyticity and unique continuation for semilinear conservative PDEs

We review some recent results in which we develop a new method for proving global unique continuation for some conservative PDEs. The main tool is to prove some global propagation of analyticity. We first present some known results on the subject. Then, we sketch the abstract method we use, which relies on the property of finite determining modes. We give applications to semilinear wave, plates and Schr\''odinger equations. This note was written for the \emph{Proceedings of the Journ{é}es EDP 2025}.

math.AP↗

arXiv2025

Stabilization and control of the nonlinear plate equation

In this article we prove semiglobal stabilization and exact controllability results for nonlinear plate equations with hinged boundary conditions and analytic nonlinearity. These results hold when the damping or control is localized in a region where observability for the linear Schrödinger equation is known to hold. At the core of these results lies a new unique continuation property for the nonlinear plate equation, which significantly relaxes the geometric conditions required for such property to hold. This property is obtained by combining recent results on propagation of analyticity in time and unique continuation for linear plate operators. More broadly, our approach exploits the linear observability of the plate equation to establish both stabilization and control results. First, we prove exponential decay of the nonlinear energy under a defocusing assumption on the nonlinearity. Second, under a weaker asymptotic assumption on the nonlinearity, we prove semiglobal exact control by analyzing control properties inside the compact attractor provided by the dynamics of the damped equation.

math.AP↗

arXiv2025

Unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schrödinger equations with analytic nonlinearity from a region $ω$ where the Geometric Control Condition holds. Our approach refines a recent technique introduced by Laurent and the author, which combines control theory techniques and Galerkin approximation, to propagate analyticity in time from a zone where observability holds. As a main consequence, we obtain unique continuation for subcritical semilinear Schrödinger equations on compact manifolds of dimension $2$ and $3$ when the solution is assumed to vanish on $ω$. Furthermore, semiglobal control and stabilization follow only under the Geometric Control Condition on the observation zone. In particular, this answers in the affirmative an open question of Dehman, Gérard, and Lebeau from $2006$ for the nonlinear case.

math.AP↗

arXiv2024

Global propagation of analyticity and unique continuation for semilinear waves

In this article, we develop a new method to prove both global propagation of analyticity and unique continuation in finite time for solutions of semilinear wave-type equations with analytic nonlinearity. It combines control theory techniques and Galerkin approximation, inspired by Hale-Raugel, to prove that analyticity in time can be propagated for the nonlinear equation from a zone where linear observability holds towards the full space.For semilinear wave equations with Dirichlet boundary condition on a bounded domain, this implies that analyticity can be propagated to the entire domain from a subset $ω$ that satisfies the geometric control condition. It also implies the unique continuation when the solution is assumed to be zero on $ω$. When the nonlinearity is assumed to be subcritical and defocusing, we also obtain observability estimates in the optimal time of the geometric control condition.For semilinear plate equations, similar propagation of analyticity is achieved by assuming the controllability of the linear Schr\''odinger equation.

math.AP↗