Exact controllability and stabilizability of the Kortewegde Vries equation
Authors:
David L. Russell and BingYu Zhang
Journal:
Trans. Amer. Math. Soc. 348 (1996), 36433672
MSC (1991):
Primary 35K60, 93C20
MathSciNet review:
1360229
Fulltext PDF Free Access
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Abstract: In this paper, we consider distributed control of the system described by the Kortewegde 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 24061123
Email:
russell@math.vt.edu
BingYu Zhang
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221
Email:
bzhang@math.uc.edu
DOI:
http://dx.doi.org/10.1090/S0002994796016728
PII:
S 00029947(96)016728
Received by editor(s):
May 12, 1994
Additional Notes:
Supported in part by NSF Grant DMS9402838. Reproduction in whole or in part is permitted for U.S. Government purposes.
Article copyright:
© Copyright 1996 American Mathematical Society
