Weak damping for the Korteweg-de Vries equation

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 welcom

The fractional Schr\"odinger equation on compact manifolds: Global controllability results

The goal of this work is to prove global controllability and stabilization properties for the fractional Schr\"odinger equation on $d$-dimensional compact Riemannian manifolds without boundary $(M,g)$. To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of singularities results which together with Strichartz type estimates and unique continuation property help us to achieve the main results of this work.Comment: 26 page

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 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: 28 pages. Comments are welcom

Controls insensitizing the norm of solution of a Schr\"odinger type system with mixed dispersion

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schr\"odinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schr\"odinger system of cascade type with zero order coupling which is equivalent to the insensitizing control problem. Precisely, by means of new Carleman estimates, we first obtain a null controllability result for the linearized system around zero, then the null controllability for the nonlinear case is extended using an inverse mapping theorem.Comment: 26 pages. Comments are welcom

Asymptotic behavior of Kawahara equation with memory effect

In this work, we are interested in a detailed qualitative analysis of the Kawahara equation, a model that has numerous physical motivations such as magneto-acoustic waves in a cold plasma and gravity waves on the surface of a heavy liquid. First, we design a feedback law, which combines a damping control and a finite memory term. Then, it is shown that the energy associated with this system exponentially decays.Comment: 20 pages. Comments are welcom