arXiv subjects

F. A. Gallego

Publications and source records attributed to F. A. Gallego.

7 recordsLinked to original sources

arXiv2025

Boundary Exponential Stabilization for the Linear KP-II equation without Critical Size Restrictions

In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as ``critical length." Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain $ Ω= (0, L) \times (0, L)$, $ L > 0 $. Furthermore, we examine the interaction between the drift term $ u_x $ under these constraints.

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arXiv2025

Global Stabilization for the BBM-KP equations on R2

In this paper, we present results on the energy decay of the BBM-KP equations (I and II) posed on $\R^2$ with localized damping. This model offers an alternative to the KP equations, analogous to how the regularized long-wave equation relates to the classical Korteweg-de Vries (KdV) equation. We show that the energy associated with the Cauchy problem decays exponentially when a localized dissipative mechanism is present in a subdomain. Finally, we validate the theoretical results on the exponential stabilization of solutions to the BBM-KP equations with damping through numerical experiments using a spectral-finite difference scheme.

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arXiv2025

Boundary Effects on the Controllability of Coupled KdV Systems

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.

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arXiv2022

The Well-posedness and Controllability of the Generalized Symmetric Regularized Long Wave System

The symmetric regularized long wave system (SRLW) is a model for the weakly nonlinear ion acoustic and space-charge waves, which was introduced by C. Seyler and D. Fenstermacher. In this paper, we investigated the wellposedness and controllability properties of the generalized symmetric regularized long wave system (g-SRLW) in different structures (periodic and bounded domains). Firstly, the wellposedness and the exact controllability results for both linear and nonlinear g-SRLW system posed on the one-dimensional torus are obtained under the effect of a distributed moving control. Second, we consider the g-SRLW system in a bounded interval with some Dirichlet-Neumann conditions and we show that the system is not spectrally controllable (No finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero). Although the system is not spectrally controllable, it can be shown that it is approximately controllable.

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arXiv2018

On the well posedness and large-time behavior of higher order Boussinesq system

A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the generalized higher order Boussinesq systems of Korteweg-de Vries--type posed on a interval. We design a two-parameter family of feedback laws for which the system is locally well-posed and the solutions of the linearized system are exponentially decreasing in time.

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arXiv2017

Controllability Aspects of the Korteweg-de Vries Burgers Equation on Unbounded Domains

The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in $L^2_{loc}(\R^2)$ by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in \cite{rosier2000}, the problem is reduced to prove an approximate theorem.

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arXiv2017

Decay Rates of the Solutions to the Thermoelastic Bresse System of Types I and III

In this paper, we study the energy decay for the thermoelastic Bresse system in the whole line with two different dissipative mechanism, given by heat conduction (Types I and III). We prove that the decay rate of the solutions are very slow. More precisely, we show that the solutions decay with the rate of $(1+t)^{-\frac{1}{8}}$ in the $L^2$-norm, whenever the initial data belongs to $L^1(R) \cap H^{s}(R)$ for a suitable $s$. The wave speeds of propagation have influence on the decay rate with respect to the regularity of the initial data. This phenomenon is known as \textit{regularity-loss}. The main tool used to prove our results is the energy method in the Fourier space.

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