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 | References | Similar Articles | Additional Information

Abstract: In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation

on the interval , with periodic boundary conditions

where the distributed control is restricted so that the ``volume'' 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 , it is shown that the system is globally exactly controllable, i.e., for given and functions , with the same ``volume'', one can alway find a control so that the system (i)--(ii) has a solution satisfying

If the control is allowed to act on only a small subset of the domain , then the same result still holds if the initial and terminal states, and , 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 . 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:
https://doi.org/10.1090/S0002-9947-96-01672-8

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