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

DOI:
https://doi.org/10.1090/S0002-9947-96-01672-8

MathSciNet review:
1360229

Full-text PDF

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.

**1.**J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation,*Philos. Trans. Roy. Soc. London A*,**278**(1975), 555 -- 601. MR**52:6219****2.**J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part I: Schrödinger equations,*Geometric and Functional Analysis***3**(1993), 107 -- 156. MR**95d:35160a****3.**J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part II: the KdV equation,*Geometric and Functional Analysis*,**3**(1993), 209 -- 262. MR**95d:35160b****4.**A. E. Ingham, Some trigonometrical inequalities with application to the theory of series,*Math. A.,***41**(1936), 367 -- 379.**5.**T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations,*Advances in Mathematics Supplementary Studies, Studies in Applied Math.*,**8**(1983), 93 -- 128. MR**86f:35160****6.**C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,*Duke Math. J.***71**(1993), 1-21. MR**94g:35196****7.**V. Komornik, D. L. Russell and B. -Y. Zhang, Stabilisation de l'equation de Korteweg-de Vries , C. R. Acad. Sci. Paris, t.**312**(1991), 841 -- 843. MR**92b:35134****8.**V. Komornik, D. L. Russell and B. -Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, submitted to*J. Differential Equations*.**9.**R. M. Miura, The Korteweg-de Vries equation: a survey of results,*SIAM Review*,**18**(1976), 412 -- 459. MR**53:8689**; MR**57:6908****10.**R. M. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equations and generalizations II: Existence of conservation laws and constants of motion,*J. Math. Physics*,**9**(1968), 1204 -- 1209. MR**40:6042b****11.**A. Pazy,*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, 1983. MR**85g:47061****12.**Lord Rayleigh, On waves,*Phil. Mag.*,**1**(1876), 257 -- 279.**13.**D. L. Russell, Computational study of the Korteweg-de Vries equation with localized control action,*Distributed Parameter Control Systems: New Trends and Applications*, G. Chen, E. B. Lee, W. Littman, and L. Markus, eds., Lecture Notes in Pure and Appl. Math., vol. 128, Marcel Dekker, New York, 1991, 195--203. MR**92c:65113****14.**D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,*SIAM Rev.,***20**(1978), 639 -- 739. MR**80c:93032****15.**D. L. Russell and B. -Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain,*SIAM J. Control and Optimization*,**31**(1993), 659 -- 676. MR**94g:93018****16.**D. L. Russell and B. -Y. Zhang, Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation,*J. Math. Anal. Appl.***190**(1995), 449--488. MR**95k:35180****17.**J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation,*Israel J. Math.*,**24**(1976), 78 -- 87. MR**56:12676****18.**R. Temam, Sur un problème non linéaire,*J. Math. Pures Appl.*,**48**(1969), 159 -- 172. MR**41:5799****19.**B. -Y. Zhang, Some results for nonlinear dispersive wave equations with applications to control,*Ph. D thesis*, University of Wisconsin-Madison, 1990.**20.**B. -Y. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain,*Differential and Integral Equations***8**(1995), 1191--1204. MR**96a:35182**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
35K60,
93C20

Retrieve articles in all journals with MSC (1991): 35K60, 93C20

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