Exact controllability and stabilizability of the Korteweg-de Vries equation期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

Exact controllability and stabilizability of the Korteweg-de Vries equation

Authors:	David L Russell Bing-Yu Zhang

Institution:	Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-123 ; Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221

Abstract:	In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation $\begin{equation}% \partial _t % u + % u % \partial _x u + % \partial _x^3 u = f \tag {i}\label {star} \end{equation}$ on the interval $0\leq x\leq 2\pi , \, t\geq 0$ , with periodic boundary conditions $\begin{equation}\partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii}\label {2star} \end{equation}$ where the distributed control $f\equiv f(x,t)$ is restricted so that the ``volume' $\int ^{2\pi }_0 u(x,t) dx$ of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control is allowed to act on the whole spatial domain $(0,2\pi )$ , it is shown that the system is globally exactly controllable, i.e., for given and functions $\phi (x)$ , $\psi (x)$ with the same ``volume', one can alway find a control so that the system (i)--(ii) has a solution satisfying $\begin{displaymath}u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .\end{displaymath}$ If the control is allowed to act on only a small subset of the domain $(0,2\pi )$ , then the same result still holds if the initial and terminal states, $\psi$ and $\phi$ , have small ``amplitude' in a certain sense. In the case of closed loop control, the distributed control is assumed to be generated by a linear feedback law conserving the ``volume' while monotonically reducing $\int ^{2\pi }_0 u(x,t)^2 dx$ . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

Keywords:

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏