arXiv subjects

Luis P. Yapu

Publications and source records attributed to Luis P. Yapu.

13 recordsLinked to original sources

arXiv2026

Hierarchical local null controllability for a non-degenerate quasi-linear parabolic system

In this manuscript, we are concerned with the hierarchical control of a non-degenerate quasi-linear parabolic system, posed on a bounded interval. We use the Stackelberg-Nash strategy with a single control called leader and two controls called followers. First, we study the controllability for the linear system, where we will demonstrate a Carleman inequality associated with the adjoint problem. Then, using the results obtained for the linear case and applying Liusternik's Inverse Function Theorem, we obtain local null controllability for the quasi-linear problem.

math.AP↗

arXiv2026

Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry

We study statistical manifolds $(M,g,\nabla,\nabla^{*})$ endowed with a nonzero $(1,1)$-tensor field $Θ$ satisfying the gauge equation \[ \nabla_X(ΘY)=Θ(\nabla_X^{*}Y). \] We first characterize this condition in terms of the statistical difference tensor \[ K=\nabla-\nabla^{g}. \] When $Θ$ is parallel with respect to the Levi-Civita connection, the gauge equation is equivalent to \[ K_XΘ=-ΘK_X. \] We further show that $Θ$ intertwines the parallel transports of the dual connections. Consequently, its rank is constant on every connected component, and $\kerΘ$ and $\operatorname{Im}Θ$ determine smooth integrable distributions. If, in addition, \[ TM=\kerΘ\overset{\perp}{\oplus}\operatorname{Im}Θ, \] we establish a local product decomposition of the statistical structure. We then study submanifolds carrying $Θ$-invariant and $Θ$-anti-invariant distributions and derive the tangential and normal components of the ambient gauge equation in terms of the second fundamental forms and shape operators of the dual statistical connections. We also obtain curvature-intertwining consequences and present a non-totally-geodesic example illustrating the submanifold identities.

math.DG↗

arXiv2026

Decay of solutions and bilinear control to trajectories of a 1D degenerate parabolic system

This paper is concerned with the decay of solutions and the analysis of the local and global exact controllability to trajectories for a class of one-dimensional nonlinear parabolic systems with weakly degenerate diffusion coefficients. The system is controlled through the coefficient of the reaction term. Our approach relies on a well-known local inversion method combined with suitable a priori estimates specifically adapted to the degenerate setting.

math.AP↗

arXiv2026

Bilinear control to trajectories of 1D degenerate parabolic equations in moving domains

In this paper, we are concerned with local controllability properties of degenerate parabolic equations in bounded domains that evolve in time. More precisely, we deal with the exact controllability to a positive trajectory of a one-dimensional semilinear degenerate equation governed via the coefficient of the reaction term. We apply a well-known local inversion method combined with some appropriate specific estimates.

math.AP↗

arXiv2026

Nash-Stackelberg controllability for coupled systems of degenerate equations in non-cylindrical domains

In this paper we investigate the Hierarchical null controllability of a coupled degenerate semilinear parabolic equation in domains which are moving in time. We show the local null controllability of the semilinear system using Liusternik's inverse function theorem. Nevertheless, the main difficulty is to adapt a Carleman estimate for the controllability of the linearized otimality system, using a Carleman inequality for degenerate non-autonomous equation obtanied by the authors previously.

math.AP↗

arXiv2025

Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms

This paper extends our previous controllability results for a class of coupled linear parabolic systems with nonlocal interactions, motivated by applications in finance such as generalized Black--Scholes models. We establish local null controllability at a fixed time T>0 for a class of semilinear, nonlocally coupled systems driven by a single internal control acting on one component. The proof combines Kakutani's fixed-point theorem with a controllability/observability estimate for the associated linearized dynamics. In addition, we obtain controllability for a broader class of linear systems than those considered in the first article. The paper concludes with remarks on boundary controllability within the same nonlocal framework and with perspectives for future research.

math.AP↗

arXiv2025

Existence, Stability and Controllability of the parabolic-parabolic thermistor model

In this article we establish the well-posedness, energy estimates, stability, and local null controllability for the thermistor system modeled by a parabolic-parabolic system using a control force acting on just one equation of the system. The proof of the controllability is based on appropriate Carleman estimates and Liusternik's inverse function theorem to obtain the local controllability of the nonlinear system. The coupling of the system happens both in the terms of order zero and one, which requires the use of a special Carleman estimate for the system.

math.AP↗

arXiv2025

Local controllability of free boundary three-dimensional semilinear radial parabolic equations

We prove that a free boundary semilinear heat equation with Stefan boundary condition and radially symmetric data is locally null controllable. The strategy involves reducing the problem to the corresponding one-dimensional formulation and adapting a Carleman inequality in that setting. The local null controllability of the free-boundary problem is then established via the Schauder fixed-point theorem. To the best of our knowledge, this is the first controllability result for this problem with Stefan boundary condition in more than one spatial dimension.

math.AP↗

arXiv2025

Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains

In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a semilinear parabolic equation in one-dimension defined in a non-cylindrical domain where the diffusion coefficient degenerates at one point of the boundary. The linearized degenerated system is treated using a Carleman inequality for degenerated non-autonomous systems proved by the autors in [19] and the local controllability of the semilinear system is obtained using Liusterniks inverse function theorem.

math.AP↗

arXiv2025

Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient

In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called \textit{leader} and two controls called \textit{followers}. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, Límaco and Viana \cite{DemarqueLimacoViana_deg_sys2020} and the local controllability of the non-linear system is obtained using Liusternik's inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem.

math.OC↗

arXiv2025

Controllability of a system of non-autonomous degenerate coupled parabolic equations

We prove a Carleman estimate for a one-dimensional parabolic equation which degenerates at one extremity of the domain and has a bounded, time dependent coefficient multiplying the diffusion term. Then we use the estimate to show the null controllability of a coupled system characterized by this form of diffusion operator and bounded coefficients.

math.AP↗

arXiv2024

Scattering for the generalized Hartree equation with a potential

We consider the focusing generalized Hartree equation in $H^1(\R^3)$ with a potential, \begin{equation*} iu_t + Δu - V(x)u + (I_γ\ast |u|^p )|u|^{p-2} u=0, \end{equation*} where $I_γ= \frac{1}{|x|^{3-γ}}$, $p \geq 2$ and $γ< 3$. In this paper, we prove scattering for the generalized Hartree equation with a potential in the intercritical case assuming radial initial data. The novelty of our approach lies in the use of a general mass-potential condition, incorporating the potential V, which extends the standard mass-energy framework. To this end, we employ a simplified method inspired by Dodson and Murphy \cite{Dod-Mur}, based on Tao's scattering criteria and Morawetz estimates. This approach provides a more straightforward proof of scattering compared to the traditional concentration-compactness/rigidity method of Kenig and Merle \cite{KENIG}.

math.AP↗

arXiv2019

Conserved quantities and Hamiltonization of nonholonomic systems

This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals). In appropriate coordinates, this 2-form is shown to agree with the one previously introduced locally in [34]. We use our coordinate-free viewpoint to study various geometric features of the reduced brackets. We apply our formulas to obtain a new geometric proof of the hamiltonization of a homogeneous ball rolling without sliding in the interior side of a convex surface of revolution using our formulas.

math-ph↗