publications

2025

2025

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

   Cristóbal Loyola

   Submitted, 2025

   Abs arXiv PDF

   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 Bukhgeim-Klibanov method, when some further assumptions on the interface are made. We conclude with some remarks on the boundary observability.

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

   Cristóbal Loyola

   Submitted, 2025

   Abs arXiv PDF

   In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schrödinger equations with analytic nonlinearity from a region ωwhere 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.

3. Stabilization and control of the nonlinear plate equation

   Cristóbal Loyola

   Submitted, 2025

   Abs arXiv PDF

   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.

2024

2024

1. Global propagation of analyticity and unique continuation for semilinear waves

   Camille Laurent, and Cristóbal Loyola

   Submitted, 2024

   Abs arXiv PDF

   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 ωthat 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 Schrödinger equation.

2023

2023

1. An explicit time for the uniform null controllability of a linear Korteweg–de Vries equation

   Nicolás Carreño, and Cristóbal Loyola

   J. Evol. Equ., 2023

   Abs PDF

   In this paper, we consider a linear Korteweg-de Vries equation posed in a bounded interval and study the time dependency with respect to the interval length and the transport coefficient, for which the uniform null controllability holds as the dispersion coefficient goes to zero. We consider two cases of boundary controls. First, only one control on the left end of the interval, and then, two controls acting on the right. The strategy is based on the combination of an exponential dissipation inequality and suitable Carleman estimates for each case.