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This article addresses the stabilizability of a perturbed quintic defocusing Schr\"odinger 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 Schr\"odinger 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., Comment: This version has been submitted with minor revisions and an updated bibliography. Feedback is highly appreciated. 56 pages. arXiv admin note: text overlap with arXiv:1004.2768 by other authors

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 achieved by introducing a new concept known as \textit{operational controllability}, which provides key insights for establishing exact controllability results for the KdV equation. This approach allows explicitly characterizing both the control input and the controllable solutions. Furthermore, this concept holds significant potential for application to a wide range of nonlinear dispersive equations on the half-line and in bounded intervals., Comment: 37 pages. Comments are welcome

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 the spatial domain length avoids certain specific values, which is 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., Comment: Comments are welcome. 20 pages

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., Comment: 30 pages. Comments are welcome

This article deals with the $H^{1}$--level local null controllability for the defocusing critical nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$. Firstly, we show the problem under consideration to be well-posed using Strichartz estimates. Moreover, through the Hilbert uniqueness method, we prove the linear Schr\"{o}dinger equation to be controllable. Finally, we use a perturbation argument and show local controllability for the critical nonlinear Schr\"{o}dinger equation., Comment: 20 pages. Several typos were removed. Comments are welcome

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., Comment: To appear Nonlinear Analysis: Real World Applications

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.

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 M\"obius transformations., Comment: 20 pages - comments are welcome

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., Comment: 20

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., Comment: 32 pages. Comments are welcome!

In this work we are concerned with solutions to the linear Schr\"odinger type system with mixed dispersion, the so-called biharmonic Schr\"odinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure $\mathcal{G}$ for $T>T_{min}$, with $$T_{min}=\sqrt{ \frac{ \overline{L} (L^2+\pi^2)}{\pi^2\varepsilon(1- \overline{L} \varepsilon)}},$$ when the couplings and the controls appear only on the Neumann boundary conditions., Comment: 15 pages, 1 figure, comments are welcome

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., Comment: 38 pages, 2 figues

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 Schr\"odinger 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., Comment: To appear on Nonlinear Differential Equations and Applications (NoDEA) - 28 pages

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 $\omega\subset\mathbb{T}$ 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., Comment: 36 pages - All the previous results are correct, but this paper is entirely reorganized and some results are rephrased or deleted. Comments are welcome

For more than 20 years, the Korteweg-de Vries equation has been intensively explored from the mathematical point of view. Regarding control theory, when adding an internal force term in this equation, it is well known that the Korteweg-de Vries equation is exponentially stable in a bounded domain. In this work, we propose a weak forcing mechanism, with a lower cost than that already existing in the literature, to achieve the result of the global exponential stability to the Korteweg-de Vries equation., Comment: 19 pages. comments are welcome

This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schr\"odinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic cubic nonlinear Schr\"odinger equation on star graphs are also discussed., Comment: 26 pages, 2 figures

In the last 40 years the study of initial boundary value problem for the Korteweg-de Vries equation has had the attention of researchers from various research fields. In this note we present a review of the main results about this topic and also introduce interesting open problems which still requires attention from the mathematical point of view., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1703.08154

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability., Comment: 27 pages

We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.

This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet--Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the $L^2(0,L)$--norm and the decay rate can be tuned to be as large as desired if the initial data is small enough., Comment: 21 pages

This article is dedicated to improve the controllability results obtained by Cerpa et al. in Commun. Contemp. Math 13 (2011) and by Micu et al. in Commun. Contemp. Math 11 (5) (2009) for a nonlinear coupled system of two Korteweg-de Vries (KdV) equations posed on a bounded interval. Initially, in Micu et al., the authors proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in Cerpa et al., two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T. Here, we use the ideas contained in Capistrano-Filho et al. (arXiv 1508.07525) to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the nonlinear system, i. e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T. In both cases, the control spaces are sharp due a technical lemma which reveals a hidden regularity for the solution of the adjoint system., Comment: arXiv admin note: text overlap with arXiv:1602.00742

In this paper we study the boundary controllability of the Gear-Grimshaw system posed on a finite domain $(0,L)$, with Neumann boundary conditions: \begin{equation} \label{abs} \begin{cases} u_t + uu_x+u_{xxx} + a v_{xxx} + a_1vv_x+a_2 (uv)_x =0, & \text{in} \,\, (0,L)\times (0,T), c v_t +rv_x +vv_x+abu_{xxx} +v_{xxx}+a_2buu_x+a_1b(uv)_x =0, & \text{in} \,\, (0,L)\times (0,T), u_{xx}(0,t)=h_0(t),\,\,u_x(L,t)=h_1(t),\,\,u_{xx}(L,t)=h_2(t), & \text{in} \,\, (0,T), v_{xx}(0,t)=g_0(t),\,\,v_x(L,t)=g_1(t),\,\,v_{xx}(L,t)=g_2(t), & \text{in} \,\, (0,T), u(x,0)= u^0(x), \quad v(x,0)= v^0(x), & \text{in} \,\, (0,L).\nonumber \end{cases} \end{equation} We first prove that the corresponding linearized system around the origin is exactly controllable in $(L^2(0,L))^2$ when $h_2(t)=g_2(t)=0$. In this case, the exact controllability property is derived for any $L>0$ with control functions $h_0, g_0\in H^{-\frac{1}{3}}(0,T)$ and $h_1, g_1\in L^2(0,T)$. If we change the position of the controls and consider $h_0(t)=h_2(t)=0$ (resp. $g_0(t)=g_2(t)=0)$ we obtain the result with control functions $g_0, g_2\in H^{-\frac{1}{3}}(0,T)$ and $h_1, g_1\in L^2(0,T)$ if and only if the length $L$ of the spatial domain $(0,L)$ belongs to a countable set. In all cases the regularity of the controls are sharp in time. If only one control act in the boundary condition, $h_0(t)=g_0(t)=h_2(t)=g_2(t)=0$ and $g_1(t)=0$ (resp. $h_1(t)=0$), the linearized system is proved to be exactly controllable for small values of the length $L$ and large time of control $T$. Finally, the nonlinear system is shown to be locally exactly controllable via the contraction mapping principle, if the associated linearized systems are exactly controllable., Comment: 30 Pages

In this paper we study boundary controllability of the Korteweg-de Vries (KdV) equation posed on a finite domain $(0,L)$ with the Neumann boundary conditions: u_t+u_x+uu_x+u_{xxx}=0 in (0,L)x(0,T), u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T), u(x,0)=u_0(x) in (0,L). We show that the associated linearized system u_t+(1+\beta)u_x+u_{xxx}=0 in (0,L)x(0,T), u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T), u(x,0)=u_0(x) in (0,L) is exactly controllable if and only if the length $L$ of the spatial domain $(0,L)$ does not equal to $-1$ or does not belong to set R_{\beta}:={\frac{2\pi}{\sqrt{3(1+\beta)}}\sqrt{k^{2}+kl+l^{2}}:k,l\in\mathbb{N}^{\ast}}\cup{\frac{k\pi}{\sqrt{1+\beta}}:k\in\mathbb{N}^{\ast}} and the nonlinear system is locally exactly controllable around a constant steady state $\beta$ if the associated linear system is exactly controllable., Comment: arXiv admin note: text overlap with arXiv:1401.6833 by other authors

This paper is devoted to the study of a nonlinear coupled system of two Korteweg-de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order., Comment: 18 pages