March  2014, 4(1): 45-99. doi: 10.3934/mcrf.2014.4.45

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

Received  January 2012 Revised  January 2013 Published  December 2013

These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in $L^2$-norm.
Citation: Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation,, Comm. in Contemp. Math., 11 (2009), 495.  doi: 10.1142/S0219199709003454.  Google Scholar

[13]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295.  doi: 10.1007/BF01211563.  Google Scholar

[14]

J.-M. Coron, On the controllability of 2-D incompressible perfect fluids,, J. Math. Pures Appl. (9), 75 (1996), 155.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (2007).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs,, Advances in Design and Control, (2008).   Google Scholar

[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.  Google Scholar

[30]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext, (2009).   Google Scholar

[31]

J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations,, Recherches en Mathématiques Appliquées, (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[47]

J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233.  doi: 10.1137/S0363012901388452.  Google Scholar

[48]

G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics, (1974).   Google Scholar

[49]

B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation,, SIAM J. Math. Anal., 23 (1992), 55.  doi: 10.1137/0523004.  Google Scholar

[50]

B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Control Optim., 37 (1999), 543.  doi: 10.1137/S0363012997327501.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation,, Comm. in Contemp. Math., 11 (2009), 495.  doi: 10.1142/S0219199709003454.  Google Scholar

[13]

J.-M. Coron, Global asymptotic stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295.  doi: 10.1007/BF01211563.  Google Scholar

[14]

J.-M. Coron, On the controllability of 2-D incompressible perfect fluids,, J. Math. Pures Appl. (9), 75 (1996), 155.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J.-M. Coron, Control and Nonlinearity,, Mathematical Surveys and Monographs, (2007).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs,, Advances in Design and Control, (2008).   Google Scholar

[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.  Google Scholar

[30]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext, (2009).   Google Scholar

[31]

J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations,, Recherches en Mathématiques Appliquées, (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

T. Tao, Nonlinear dispersive equations. Local and global analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[47]

J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233.  doi: 10.1137/S0363012901388452.  Google Scholar

[48]

G. B. Whitham, Linear and Nonlinear Waves,, Pure and Applied Mathematics, (1974).   Google Scholar

[49]

B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation,, SIAM J. Math. Anal., 23 (1992), 55.  doi: 10.1137/0523004.  Google Scholar

[50]

B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Control Optim., 37 (1999), 543.  doi: 10.1137/S0363012997327501.  Google Scholar

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