This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Crépeau established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is {\it not} small-time locally controllable.
The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathematical theories are built on this totally unknown constant. We introduce a constructive method that gives the quantitative value of this constant.
In recent years, controllability problems for dispersive systems have been extensively studied. This work is dedicated to proving a new type of controllability for a dispersive fifth order equation that models water waves, what we will now call the overdetermination control problem. Precisely, we are able to find a control acting at the boundary that guarantees that the solutions of the problem under consideration satisfy an overdetermination integral condition. In addition, when we make the control act internally in the system, instead of the boundary, we are also able to prove that this cond…
We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial d…
Stability of a Korteweg--de Vries equation close to critical lengths
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
In this article, we prove that small localized data yield solutions to Higher order Korteweg-de Vries type equation with scattering-supercritical nonlinearity have linear dispersive decay in only a finite length of time. The proof is done by using space-time resonance method and analyzing the oscillatory integrals on the Fourier side.
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 go…
Critical lengths for the linear Kadomtsev-Petviashvili II equation
Universidade Federal de Pernambuco (UFPE)（巴西联邦伯南布哥大学）；Universidad Nacional de Colombia (UNAL)（哥伦比亚国立大学）；University of Dubrovnik（杜布罗夫尼克大学）；Universidad de Chile（智利大学）
本文研究了二维卡多莫特塞夫-彼得萨夫斯基 II 方程的临界长度问题，通过推导可观察性不等式，实现了线性系统的精确边界可控性和边界指数稳定化。
