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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Exact controllability and stabilizability
of the Korteweg-de Vries equation


Authors: David L. Russell and Bing-Yu Zhang
Journal: Trans. Amer. Math. Soc. 348 (1996), 3643-3672
MSC (1991): Primary 35K60, 93C20
MathSciNet review: 1360229
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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 $f$ 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 $T> 0$ and functions $\phi (x)$, $\psi (x)$ with the same ``volume'', one can alway find a control $f$ so that the system (i)--(ii) has a solution $u(x,t)$ satisfying

\begin{displaymath}u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .\end{displaymath}

If the control $f$ 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 $f$ 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.


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Additional Information

David L. Russell
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-123
Email: russell@math.vt.edu

Bing-Yu Zhang
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221
Email: bzhang@math.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01672-8
PII: S 0002-9947(96)01672-8
Received by editor(s): May 12, 1994
Additional Notes: Supported in part by NSF Grant DMS-9402838. Reproduction in whole or in part is permitted for U.S. Government purposes.
Article copyright: © Copyright 1996 American Mathematical Society