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Control of a Korteweg-de Vries equation: A tutorial
1. | Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile |
References:
[1] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.
doi: 10.1137/0330055. |
[2] |
L. Baudouin, E. Cerpa, E. Crépeau and A. Mercado, On the determination of the principal coefficient from boundary measurements in a KdV equation,, J. Inverse Ill-Posed Probl., ().
doi: 10.1515/jip-2013-0015. |
[3] |
K. Beauchard, Local controllability of a 1-D Schrödinger equation,, J. Math. Pures Appl. (9), 84 (2005), 851.
doi: 10.1016/j.matpur.2005.02.005. |
[4] |
K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well,, J. Funct. Anal., 232 (2006), 328.
doi: 10.1016/j.jfa.2005.03.021. |
[5] |
J. Bona and R. Winther, The Korteweg-de Vries equation, posed in a quarter-plane,, SIAM J. Math. Anal., 14 (1983), 1056.
doi: 10.1137/0514085. |
[6] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983).
|
[7] |
E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain,, SIAM J. Control Optim., 46 (2007), 877.
doi: 10.1137/06065369X. |
[8] |
E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition,, IEEE Trans. Automat. Control, 58 (2013), 1688.
doi: 10.1109/TAC.2013.2241479. |
[9] |
E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457.
doi: 10.1016/j.anihpc.2007.11.003. |
[10] |
E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655.
doi: 10.3934/dcdsb.2009.11.655. |
[11] |
E. Cerpa, I. Rivas and B.-Y. Zhang, Boundary controllability of the Korteweg-de Vries equation on a bounded domain,, SIAM J. Control Optim., 51 (2013), 2976.
doi: 10.1137/120891721. |
[12] |
M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation,, Comm. in Contemp. Math., 11 (2009), 495.
doi: 10.1142/S0219199709003454. |
[13] |
J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295.
doi: 10.1007/BF01211563. |
[14] |
J.-M. Coron, On the controllability of 2-D incompressible perfect fluids,, J. Math. Pures Appl. (9), 75 (1996), 155.
|
[15] |
J.-M. Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations,, ESAIM Control Optim. Calc. Var., 8 (2002), 513.
doi: 10.1051/cocv:2002050. |
[16] |
J.-M. Coron, On the small time local controllability of a quantum particle in a moving one-dimensional infinite square potential well,, C. R. Acad. Sci. Paris, 342 (2006), 103.
doi: 10.1016/j.crma.2005.11.004. |
[17] |
J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (2007).
|
[18] |
J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths,, J. Eur. Math. Soc., 6 (2004), 367.
|
[19] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM J. Control Optim., 43 (2004), 549.
doi: 10.1137/S036301290342471X. |
[20] |
E. Crépeau, Exact controllability of the Korteweg-de Vries equation around a non-trivial stationary solution,, Internat. J. Control, 74 (2001), 1096.
doi: 10.1080/00207170110052202. |
[21] |
E. Crépeau, Contrôlabilité Exacte d'Équations Dispersives Issues de la Mécanique,, Ph.D thesis, (2002). Google Scholar |
[22] |
L. Escauriaza, C. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, J. Funct. Anal., 244 (2007), 504.
doi: 10.1016/j.jfa.2006.11.004. |
[23] |
A. V. Faminskii, Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation,, Differential Integral Equations, 20 (2007), 601.
|
[24] |
O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit,, Asymptot. Anal., 60 (2008), 61.
|
[25] |
O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition,, Systems Control Lett., 59 (2010), 390.
doi: 10.1016/j.sysconle.2010.05.001. |
[26] |
C. Kenig, G. Ponce and L. Vega, On the unique continuation of solutions to the generalized KdV equation,, Math. Res. Lett., 10 (2003), 833.
doi: 10.4310/MRL.2003.v10.n6.a10. |
[27] |
D. J. Korteweg and G. de Vries, 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), 422.
doi: 10.1080/14786449508620739. |
[28] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs,, Advances in Design and Control, (2008).
|
[29] |
F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries with localized damping,, Proc. Amer. Math. Soc., 135 (2007), 1515.
doi: 10.1090/S0002-9939-07-08810-7. |
[30] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext, (2009).
|
[31] |
J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations,, Recherches en Mathématiques Appliquées, (1988).
|
[32] |
C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping,, Math. Methods Appl. Sci., 30 (2007), 1419.
doi: 10.1002/mma.847. |
[33] |
A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473.
doi: 10.1051/cocv:2005015. |
[34] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).
doi: 10.1007/978-1-4612-5561-1. |
[35] |
G. Perla Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping,, Quart. Appl. Math., 60 (2002), 111.
|
[36] |
L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33.
doi: 10.1051/cocv:1997102. |
[37] |
L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346.
doi: 10.1051/cocv:2004012. |
[38] |
L. Rosier and B.-Y Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain,, SIAM J. Control Optim., 45 (2006), 927.
doi: 10.1137/050631409. |
[39] |
L. Rosier and B.-Y Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses,, J. Syst. Sci. Complex., 22 (2009), 647.
doi: 10.1007/s11424-009-9194-2. |
[40] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639.
doi: 10.1137/1020095. |
[41] |
D. L. Russell and B.-Y Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain,, SIAM J. Control Optim., 31 (1993), 659.
doi: 10.1137/0331030. |
[42] |
D. L. Russell and B.-Y Zhang, Smoothing and decay properties of the Korteweg-de Vries equation on a periodic domain with point dissipation,, J. Math. Anal. Appl., 190 (1995), 449.
doi: 10.1006/jmaa.1995.1087. |
[43] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equation,, J. Differential Equations, 66 (1987), 118.
doi: 10.1016/0022-0396(87)90043-X. |
[44] |
J. Simon, Compact sets in the space $L^p(0,T,B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.
doi: 10.1007/BF01762360. |
[45] |
S. M. Sun, The Korteweg-de Vries equation on a periodic domain with singular-point dissipation,, SIAM J. Control Optim., 34 (1996), 892.
doi: 10.1137/S0363012994269491. |
[46] |
T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[47] |
J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233.
doi: 10.1137/S0363012901388452. |
[48] |
G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics, (1974).
|
[49] |
B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation,, SIAM J. Math. Anal., 23 (1992), 55.
doi: 10.1137/0523004. |
[50] |
B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Control Optim., 37 (1999), 543.
doi: 10.1137/S0363012997327501. |
show all references
References:
[1] |
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.
doi: 10.1137/0330055. |
[2] |
L. Baudouin, E. Cerpa, E. Crépeau and A. Mercado, On the determination of the principal coefficient from boundary measurements in a KdV equation,, J. Inverse Ill-Posed Probl., ().
doi: 10.1515/jip-2013-0015. |
[3] |
K. Beauchard, Local controllability of a 1-D Schrödinger equation,, J. Math. Pures Appl. (9), 84 (2005), 851.
doi: 10.1016/j.matpur.2005.02.005. |
[4] |
K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well,, J. Funct. Anal., 232 (2006), 328.
doi: 10.1016/j.jfa.2005.03.021. |
[5] |
J. Bona and R. Winther, The Korteweg-de Vries equation, posed in a quarter-plane,, SIAM J. Math. Anal., 14 (1983), 1056.
doi: 10.1137/0514085. |
[6] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983).
|
[7] |
E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain,, SIAM J. Control Optim., 46 (2007), 877.
doi: 10.1137/06065369X. |
[8] |
E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition,, IEEE Trans. Automat. Control, 58 (2013), 1688.
doi: 10.1109/TAC.2013.2241479. |
[9] |
E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457.
doi: 10.1016/j.anihpc.2007.11.003. |
[10] |
E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655.
doi: 10.3934/dcdsb.2009.11.655. |
[11] |
E. Cerpa, I. Rivas and B.-Y. Zhang, Boundary controllability of the Korteweg-de Vries equation on a bounded domain,, SIAM J. Control Optim., 51 (2013), 2976.
doi: 10.1137/120891721. |
[12] |
M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation,, Comm. in Contemp. Math., 11 (2009), 495.
doi: 10.1142/S0219199709003454. |
[13] |
J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295.
doi: 10.1007/BF01211563. |
[14] |
J.-M. Coron, On the controllability of 2-D incompressible perfect fluids,, J. Math. Pures Appl. (9), 75 (1996), 155.
|
[15] |
J.-M. Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations,, ESAIM Control Optim. Calc. Var., 8 (2002), 513.
doi: 10.1051/cocv:2002050. |
[16] |
J.-M. Coron, On the small time local controllability of a quantum particle in a moving one-dimensional infinite square potential well,, C. R. Acad. Sci. Paris, 342 (2006), 103.
doi: 10.1016/j.crma.2005.11.004. |
[17] |
J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (2007).
|
[18] |
J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths,, J. Eur. Math. Soc., 6 (2004), 367.
|
[19] |
J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM J. Control Optim., 43 (2004), 549.
doi: 10.1137/S036301290342471X. |
[20] |
E. Crépeau, Exact controllability of the Korteweg-de Vries equation around a non-trivial stationary solution,, Internat. J. Control, 74 (2001), 1096.
doi: 10.1080/00207170110052202. |
[21] |
E. Crépeau, Contrôlabilité Exacte d'Équations Dispersives Issues de la Mécanique,, Ph.D thesis, (2002). Google Scholar |
[22] |
L. Escauriaza, C. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, J. Funct. Anal., 244 (2007), 504.
doi: 10.1016/j.jfa.2006.11.004. |
[23] |
A. V. Faminskii, Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation,, Differential Integral Equations, 20 (2007), 601.
|
[24] |
O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit,, Asymptot. Anal., 60 (2008), 61.
|
[25] |
O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition,, Systems Control Lett., 59 (2010), 390.
doi: 10.1016/j.sysconle.2010.05.001. |
[26] |
C. Kenig, G. Ponce and L. Vega, On the unique continuation of solutions to the generalized KdV equation,, Math. Res. Lett., 10 (2003), 833.
doi: 10.4310/MRL.2003.v10.n6.a10. |
[27] |
D. J. Korteweg and G. de Vries, 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), 422.
doi: 10.1080/14786449508620739. |
[28] |
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs,, Advances in Design and Control, (2008).
|
[29] |
F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries with localized damping,, Proc. Amer. Math. Soc., 135 (2007), 1515.
doi: 10.1090/S0002-9939-07-08810-7. |
[30] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext, (2009).
|
[31] |
J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations,, Recherches en Mathématiques Appliquées, (1988).
|
[32] |
C. P. Massarolo, G. P. Menzala and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping,, Math. Methods Appl. Sci., 30 (2007), 1419.
doi: 10.1002/mma.847. |
[33] |
A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473.
doi: 10.1051/cocv:2005015. |
[34] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983).
doi: 10.1007/978-1-4612-5561-1. |
[35] |
G. Perla Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping,, Quart. Appl. Math., 60 (2002), 111.
|
[36] |
L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33.
doi: 10.1051/cocv:1997102. |
[37] |
L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346.
doi: 10.1051/cocv:2004012. |
[38] |
L. Rosier and B.-Y Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain,, SIAM J. Control Optim., 45 (2006), 927.
doi: 10.1137/050631409. |
[39] |
L. Rosier and B.-Y Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses,, J. Syst. Sci. Complex., 22 (2009), 647.
doi: 10.1007/s11424-009-9194-2. |
[40] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639.
doi: 10.1137/1020095. |
[41] |
D. L. Russell and B.-Y Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain,, SIAM J. Control Optim., 31 (1993), 659.
doi: 10.1137/0331030. |
[42] |
D. L. Russell and B.-Y Zhang, Smoothing and decay properties of the Korteweg-de Vries equation on a periodic domain with point dissipation,, J. Math. Anal. Appl., 190 (1995), 449.
doi: 10.1006/jmaa.1995.1087. |
[43] |
J.-C. Saut and B. Scheurer, Unique continuation for some evolution equation,, J. Differential Equations, 66 (1987), 118.
doi: 10.1016/0022-0396(87)90043-X. |
[44] |
J. Simon, Compact sets in the space $L^p(0,T,B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.
doi: 10.1007/BF01762360. |
[45] |
S. M. Sun, The Korteweg-de Vries equation on a periodic domain with singular-point dissipation,, SIAM J. Control Optim., 34 (1996), 892.
doi: 10.1137/S0363012994269491. |
[46] |
T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[47] |
J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233.
doi: 10.1137/S0363012901388452. |
[48] |
G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics, (1974).
|
[49] |
B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation,, SIAM J. Math. Anal., 23 (1992), 55.
doi: 10.1137/0523004. |
[50] |
B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Control Optim., 37 (1999), 543.
doi: 10.1137/S0363012997327501. |
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