Control of a Korteweg-de Vries equation: A tutorial

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

Control of a Korteweg-de Vries equation: A tutorial

Eduardo Cerpa ^1,

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

Abstract
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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.

Keywords: nonlinear control., controllability, critical domain, stabilization, Korteweg-de Vries equation.

Mathematics Subject Classification: Primary: 93C20, 35Q53; Secondary: 93B05, 93D1.

Citation: Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control and 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), 1024-1065. 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., in press. doi: 10.1515/jip-2013-0015.
[3]	K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), 84 (2005), 851-956. 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), 328-389. 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), 1056-1106. 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], Masson, Paris, 1983.
[7]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899. 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), 1688-1695. 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), 457-475. 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), 655-668. 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), 2976-3010. doi: 10.1137/120891721.
[12]	M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation, Comm. in Contemp. Math., 11 (2009), 495-521. doi: 10.1142/S0219199709003454.
[13]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[14]	J.-M. Coron, On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), 75 (1996), 155-188.
[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), 513-554. 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), 103-108. doi: 10.1016/j.crma.2005.11.004.
[17]	J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 367-398.
[19]	J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569. 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), 1096-1106. doi: 10.1080/00207170110052202.
[21]	E. Crépeau, Contrôlabilité Exacte d'Équations Dispersives Issues de la Mécanique, Ph.D thesis, Université Paris-Sud, Paris, 2002.
[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), 504-535. 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), 601-642.
[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), 61-100.
[25]	O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395. 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), 833-846. 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), 422-443. doi: 10.1080/14786449508620739.
[28]	M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs, Advances in Design and Control, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.
[30]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009.
[31]	J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 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), 1419-1435. 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), 473-486. doi: 10.1051/cocv:2005015.
[34]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 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), 111-129.
[36]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. 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), 346-380. 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), 927-956. 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), 647-682. 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), 639-739. 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), 659-676. 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), 449-488. doi: 10.1006/jmaa.1995.1087.
[43]	J.-C. Saut and B. Scheurer, Unique continuation for some evolution equation, J. Differential Equations, 66 (1987), 118-139. 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), 65-96. 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), 892-912. doi: 10.1137/S0363012994269491.
[46]	T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
[47]	J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244. doi: 10.1137/S0363012901388452.
[48]	G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[49]	B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal., 23 (1992), 55-71. doi: 10.1137/0523004.
[50]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim., 37 (1999), 543-565. 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), 1024-1065. 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., in press. doi: 10.1515/jip-2013-0015.
[3]	K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), 84 (2005), 851-956. 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), 328-389. 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), 1056-1106. 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], Masson, Paris, 1983.
[7]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899. 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), 1688-1695. 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), 457-475. 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), 655-668. 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), 2976-3010. doi: 10.1137/120891721.
[12]	M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation, Comm. in Contemp. Math., 11 (2009), 495-521. doi: 10.1142/S0219199709003454.
[13]	J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[14]	J.-M. Coron, On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), 75 (1996), 155-188.
[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), 513-554. 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), 103-108. doi: 10.1016/j.crma.2005.11.004.
[17]	J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 367-398.
[19]	J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569. 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), 1096-1106. doi: 10.1080/00207170110052202.
[21]	E. Crépeau, Contrôlabilité Exacte d'Équations Dispersives Issues de la Mécanique, Ph.D thesis, Université Paris-Sud, Paris, 2002.
[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), 504-535. 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), 601-642.
[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), 61-100.
[25]	O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395. 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), 833-846. 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), 422-443. doi: 10.1080/14786449508620739.
[28]	M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs, Advances in Design and Control, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 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), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.
[30]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext, Springer, New York, 2009.
[31]	J.-L. Lions, Contrôlabilité Exacte, Perurbations et Stabilization de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 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), 1419-1435. 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), 473-486. doi: 10.1051/cocv:2005015.
[34]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 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), 111-129.
[36]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. 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), 346-380. 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), 927-956. 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), 647-682. 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), 639-739. 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), 659-676. 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), 449-488. doi: 10.1006/jmaa.1995.1087.
[43]	J.-C. Saut and B. Scheurer, Unique continuation for some evolution equation, J. Differential Equations, 66 (1987), 118-139. 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), 65-96. 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), 892-912. doi: 10.1137/S0363012994269491.
[46]	T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
[47]	J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244. doi: 10.1137/S0363012901388452.
[48]	G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[49]	B.-Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal., 23 (1992), 55-71. doi: 10.1137/0523004.
[50]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim., 37 (1999), 543-565. doi: 10.1137/S0363012997327501.

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