Exact boundary controllability of the Korteweg-de Vries equation

SIAM Journal on Control and Optimization

Volumn 37, Issue 2, 1998, Pages 543-565

  Zhang, Bing Yu ^a  

^a UNIVERSITY OF CINCINNATI   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTROL SYSTEM ANALYSIS; DISTRIBUTED PARAMETER CONTROL SYSTEMS;

BOUNDARY CONTROL; KORTEWEG-DE VRIES EQUATION;

CONTROLLABILITY;

EID: 0032201986     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0363012997327501     Document Type: Article

References (33)

1. M. J. ABLOWITZ, D. J. KAUP, A. C. NEWELL, AND H. SEGUR, The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), pp. 249-315.
2. J. L. BONA AND R. SMITH, The initial value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), pp. 555-601.
3. J. L. BONA AND B.-Y. ZHANG, The initial-value problem for the forced Korteweg-de Vries equation, Proc. Roy. Soc. Edinburgh Ser. A. 126 (1996), pp. 571-598.
4. J. BOURGAIN, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part I: Schrödinger equations, Geom. Funct. Anal., 3 (1993), pp. 107-156.
5. J. BOURGAIN, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), pp. 209-262.
6. A. COHEN, Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J., 45 (1978), pp. 149-181.
7. P. CONSTANTIN AND J.-C. SAUT, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), pp. 413-446.
8. W. CRAIG, T. KAPPELER, AND W. A. STRAUSS, Gain of regularity for equations of KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), pp. 147-186.
9. C. S. GARDNER, J. M. GREENE, M. D. KRUSKAL, AND R. M. MIURA, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), pp. 1095-1097.
10. T. KATO, On the Cauchy problem for the (generalized) Korteweg-de Vries equations, in Advances in Mathematics Supplementary Studies, Stud. Appl. Math. 8, Academic Press, New York, 1983, pp. 93-128.
11. C. E. KENIG, G. PONCE, AND L. VEGA, Well-posedness of the initial value problem for the KdV equation, J. Amer. Math. Soc., 4 (1991), pp. 323-347.
12. C. E. KENIG, G. PONCE, AND L. VEGA, Well-posedness and scattering results for the generalized Korteweg-de Vries equations via the contraction principle, Comm. Pure Appl. Math., 46 (1993), pp. 527-620.
13. 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), pp. 1-21.
14. C. E. KENIG, G. PONCE, AND L. VEGA, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), pp. 573-603.
15. V. KOMORNIK, D. L. RUSSELL, AND B.-Y. ZHANG, Stabilisation de l'équation de Korteweg-de Vries, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), pp. 841-843.
16. D. J. KORTEWEG AND G. DEVRIES, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), pp. 422-443.
17. W. LITTMAN, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), pp. 567-580.
18. W. LITTMAN AND S. TAYLOR, Smoothing evolution equations and boundary control theory. Festschrift on the occasion of the 70th birthday of Shmuel Agmon, J. Anal. Math., 59 (1992), pp. 117-131.
19. R. M. MIURA, The Korteweg-de Vries equation: A survey of results, SIAM Rev., 18 (1976), pp. 412-459.
20. A. PAZY, Semigroup of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer-Verlag, New York, 1983.
21. L. ROSIER, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESIAM Control Optim. Cal. Var., 2 (1997), pp. 33-55.
22. D. L. RUSSELL, Boundary value control of the higher-dimensional wave equation, SIAM J. Control Optim., 9 (1971), pp. 29-42.
23. D. L. RUSSELL, Boundary value control theory of the higher-dimensional wave equation part II. SIAM J. Control Optim., 9 (1971), pp. 401-419.
24. D. L. RUSSELL, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), pp. 639-739.
25. D. L. RUSSELL AND B.-Y. ZHANG, Controllability and stabilizability of the third-order linear dispersion equation of a periodic domain, SIAM J. Control Optim., 31 (1993), pp. 659-676.
26. 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), pp. 449-488.
27. D. L. RUSSELL AND B.-Y. ZHANG, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), pp. 3643-3672.
28. J. C. SAUT AND R. TEMAM, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24 (1976), pp. 78-87.
29. R. TEMAM, Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969), pp. 159-172.
30. B.-Y. ZHANG, Unique continuation for the Korteweg-De Vries equation, SIAM J. Math. Anal., 23 (1992), pp. 55-71.
31. B.-Y. ZHANG, Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values, J. Funct. Anal., 129 (1995), pp. 293-324.
32. B.-Y. ZHANG, Analyticity of solutions of the generalized Korteweg-de Vries equations with respect to their initial values, SIAM J. Math. Anal., 26 (1995), pp. 1488-1513.
33. B.-Y. ZHANG, A remark on the Cauchy problem of the periodic Korteweg-de Vries equation, Differential Integral Equations, 8 (1995), pp. 1191-1204.