arXiv subjects

Roberto de A. Capistrano-Filho

Publications and source records attributed to Roberto de A. Capistrano-Filho.

At least 19 recordsLinked to original sources

arXiv2026

Critical lengths for the linear Kadomtsev-Petviashvili II equation

The critical length phenomenon of the Korteweg-de Vries equation is well known; however, in higher dimensions, it is unknown. This work explores this property in the context of the Kadomtsev-Petviashvili equation, a two-dimensional generalization of the Korteweg-de Vries equation. Specifically, we demonstrate observability inequalities for this equation, which allow us to deduce the exact boundary controllability and boundary exponential stabilization of the linear system, provided that the spatial domain length avoids certain specific values, a direct consequence of the Paley-Wiener theorem. To the best of our knowledge, our work introduces new results by identifying a set of critical lengths for the two-dimensional Kadomtsev-Petviashvili equation.

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arXiv2026

Solitary waves for a higher order Boussinesq system: Stability and numerical experiments

In this work, we study the nonlinear orbital stability of solitary-wave solutions for a class of higher-order Boussinesq systems with Hamiltonian structure. Using variational methods and the asymptotic connection with generalized fifth-order KdV equations, we establish orbital stability results for a broad family of homogeneous and nonhomogeneous nonlinearities satisfying suitable scaling assumptions. We also perform numerical simulations to investigate the stability criterion associated with the solitary waves. The numerical results suggest that the range of wave velocities leading to orbital stability may be larger than that predicted by the theoretical analysis.

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arXiv2026

Another look at the control properties of the Korteweg-de Vries equation

This paper represents a new perspective in understanding the controllability of the Korteweg-de Vries (KdV) equation on unbounded domains. By studying the equation on both the right and left half-line with a single control input, we show that a class of solutions exists for which the KdV equation is exactly controllable. This is accomplished through the introduction of a method for explicitly characterizing controls arising from the Hilbert Uniqueness Method, referred to as operational controllability, which yields fundamental insights for proving exact controllability results for the KdV equation. This approach allows for explicitly characterizing both the control input and the controllable solutions. Furthermore, this concept holds significant potential for application to various nonlinear dispersive equations on the half-line and in bounded intervals.

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arXiv2026

Unique Continuation for Fifth-Order KP Equation and its application to control problems

We develop a framework for the fifth-order Kadomtsev--Petviashvili equation on $\mathbb{T}_x \times \mathbb{R}_y$ within a mean-zero KP-adapted Sobolev scale. A localized high-order feedback acting on the periodic variable yields a $5/2$--derivative gain in suitable space--time norms, leading to propagation of regularity and a unique continuation property for the linear dynamics. As a consequence, we derive an observability inequality for the adjoint system and establish exponential stabilization of the nonlinear closed-loop equation: for small initial data in $X_{s,0}$, $s>2$, solutions are global and decay exponentially in $X_s$. Combining observability with the Hilbert Uniqueness Method and a fixed-point argument, we obtain local exact controllability near the origin, with $L^2$ controls supported in the feedback region and cost linear in the data size. The analysis relies on a novel combination of unique continuation, frequency grouping, and the one-sided Fourier vanishing mechanism introduced for the Benjamin--Ono equation by Linares and Rosier in \textit{Trans. Amer. Math. Soc.} (2015)~\cite{LR}, here extended to the fifth-order Kadomtsev--Petviashvili equation.

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arXiv2026

Control of the Schrödinger equation in $\mathbb{R}^3$: The critical case

This article deals with the $H^{1}$--level local null controllability for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$. Firstly, we demonstrate that the problem under consideration is well-posed using Strichartz estimates. Moreover, through the Hilbert uniqueness method, we prove the linear Schrödinger equation to be controllable. Finally, we use a perturbation argument and show local controllability for the critical nonlinear Schrödinger equation.

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arXiv2026

Studies on the Rao-Nakra Sandwich Beam: Well-Posedness, Dynamics, and Controllability

In this work, we investigate the well-posedness, stabilization, and boundary controllability of a linear Rao-Nakra type sandwich beam. The system consists of three coupled equations that represent the longitudinal displacements of the outer layers and the transverse displacement of the composite beam, all of which are coupled with dynamical boundary conditions. In the first problem, time-varying interior damping and static boundary conditions with time-dependent delays are considered. Then, we establish the existence and uniqueness of solutions for the Cauchy problem associated with the damped system using semigroup theory and a classical result by Kato. Furthermore, employing a Lyapunov-based approach, we prove that the system's energy decays exponentially, despite the presence of time-varying weights and delays. In the second problem, we consider a boundary linear control system with dynamical boundary conditions, and prove its well-posedness. By deriving an observability inequality for the adjoint system and applying the Hilbert Uniqueness Method (HUM), we show that the system is null controllable. A key contribution of this work lies in handling the full three-equation coupled system, which involves significant difficulty due to the dynamic boundary conditions, resolved via appropriately constructed Lyapunov functionals and intermediate observability inequalities.

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arXiv2025

Traveling-wave solutions for a higher-order Boussinesq system: existence and numerical analysis

We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class of homogeneous and non-homogeneous nonlinearities. We establish the existence of traveling wave solutions using the variational structure of the system and the \textit{concentration-compactness} principle by P.-L. Lions, even though the nonlinearity could be non-homogeneous. For the homogeneous case, the traveling wave equations of the Boussinesq system are approximated using a spectral approach based on a Fourier basis, along with an iterative method that includes appropriate stabilizing factors to ensure convergence. In the non-homogeneous case, we apply a collocation Fourier method supplemented by Newton's iteration. Additionally, we present numerical experiments that explore cases in which the wave velocity falls outside the theoretical range of existence.

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arXiv2025

On the controllability of laminated beams with Venttsel-type boundary conditions

This paper examines the boundary controllability of a Timoshenko laminated beam system subject to Venttsel-type boundary conditions. The study focuses on a novel configuration in which three controls are applied solely at the boundary of the beam. Controllability is established by deriving an appropriate observability inequality for the corresponding adjoint system, which is then employed within the framework of the duality method in the setup of the classical Hilbert uniqueness method (HUM) to achieve the control problem. The main contribution lies in the analysis of a system comprising three beams governed by dynamic Venttsel-type boundary conditions, as introduced by Venttsel in [Theory Probab. Appl., 4 (1959)].

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arXiv2024

Stabilization of a perturbed quintic defocusing Schrödinger equation in $\mathbb{R}^{3}$

This article addresses the stabilizability of a perturbed quintic defocusing Schrödinger equation in $\mathbb{R}^{3}$ at the $H^1$--energy level, considering the influence of a damping mechanism. More specifically, we establish a profile decomposition for both linear and nonlinear systems and use them to show that, under certain conditions, the sequence of nonlinear solutions can be effectively linearized. Lastly, through microlocal analysis techniques, we prove the local exponential stabilization of the solution to the perturbed Schrödinger equation in $\mathbb{R}^{3}$ showing an observability inequality for the solution of the system under consideration, which is the key result of this work.

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arXiv2024

Control of Kawahara equation using flat outputs

In this study we focused on the linear Kawahara equation in a bounded domain, employing two boundary controls. The controllability of this system has been previously demonstrated over the past decade using the Hilbert uniqueness method which involves proving an observability inequality, in general, demonstrated via Carleman estimates. Here, we extend this understanding by achieving the exact controllability within a space of analytic functions, employing the flatness approach which is a new approach for higher-order dispersive systems.

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arXiv2024

On the boundary stabilization of the KdV-KdV system with time-dependent delay

The boundary stabilization problem of the Boussinesq KdV-KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, first, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato's approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV-KdV system decays exponentially by employing the Lyapunov method.

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arXiv2023

Control results for a model of resonant interaction between short and long capillary-gravity waves

The purpose of this article is the investigation of the global control properties of a coupled nonlinear dispersive system posed in the periodic domain $\mathbb{T}$, a system with the structure of a nonlinear Schrödinger equation and a nonlinear Korteweg-de Vries equation. Combining estimates derived from Bourgain spaces and using microlocal analysis we show that this system has global control properties. The main novelty of this work is twofold. One is that the global results for the nonlinear system are presented for the first time thanks to the propagation of singularities. The second one is that these propagation results are shown to a coupled dispersive system with two equations defined by differential operators with principal symbols of different orders.

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arXiv2022

Stabilization of the Kawahara-Kadomtsev-Petviashvili equation with time-delayed feedback

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara-Kadomtsev-Petviashvili equation are locally and globally exponentially stable. The main novelty is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

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arXiv2022

Two stability results for the Kawahara equation with a time-delayed boundary control

In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy and using the Lyapunov approach, to ensure that the unique solution of the Kawahara system exponentially decays. The second result is achieved by means of a compactness-uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the main novelty of this work is to characterize the critical set phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the Möbius transformations.

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arXiv2022

Global control aspects for long waves in nonlinear dispersive media

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two distributed controls, supported in $ω\subset\mathbb{T}$ and assumed to be generated by a linear feedback law conserving the "mass" (or "volume"), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in $H^s(\mathbb{T})$, for $s\geq0$. After that, by certain properties of Bourgain spaces we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg-de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

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arXiv2022

Massera's theorems for a higher order dispersive system

This work is devoted to present Massera-type theorems for the Kawahara system, a higher order dispersive equation, posed in a bounded domain. Precisely, thanks to some properties of the semigroup and the decays of the solutions of this equation, we are able to prove its solutions are periodic, quasi-periodic and almost periodic.

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arXiv2021

Stabilization results for delayed fifth order KdV-type equation in a bounded domain

Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients we prove that solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the Kawahara system is exponentially stable for $T> T_{\min}$. After that, introducing a more general delayed system, and by introducing suitable energies, we show using Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.

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arXiv2021

On the control issues for higher-order nonlinear dispersive equations on the circle

The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(\mathbb{T})$ for any $s \geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.

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