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October  2022, 11(5): 1745-1773. doi: 10.3934/eect.2021062

Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas-SP, Brazil

* Corresponding author

Received  August 2021 Published  October 2022 Early access  December 2021

In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in $ H_{p}^{s}(\mathbb{T}) $ with $ s\in \mathbb{R}. $ We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in $ H_{p}^{s}(\mathbb{T}) $ with $ s\in \mathbb{R}. $

Citation: Francisco J. Vielma leal, Ademir Pastor. Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain. Evolution Equations and Control Theory, 2022, 11 (5) : 1745-1773. doi: 10.3934/eect.2021062
References:
[1]

L. AbdelouhabJ. L. BonaM. Fell and J-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D., 40 (1989), 360-392.  doi: 10.1016/0167-2789(89)90050-X.

[2]

A. AnkiewiczD. J. KedzioraA. ChowduryU. Bandelow and N. Akhmediev, Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys. Rev. E., 93 (2016), 012206.  doi: 10.1103/physreve.93.012206.

[3]

J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587.  doi: 10.1002/cpa.3160320405.

[4]

R. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic Schrödinger equation, Appl. Math. Optim., 84 (2021), 103-144. 

[5]

R. Capistrano-Filho and A. Gomes, Global control aspects for long waves in nonlinear dispersive media, preprint, arXiv: 2013.00921v1.

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolutions Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.

[7]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, Amer. Math. Soc., 2007. doi: 10.1090/surv/136.

[8]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. 

[9]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schr$\ddot{o}$dinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.

[10]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[11]

C. Flores, Control and stability of the linearized dispersion-generalized Benjamin-Ono equation on a periodic domain, Math. Control Signals Systems, 30 (2018), Art. 13, 16pp. doi: 10.1007/s00498-018-0219-z.

[12]

C. FloresS. Oh and D. Smith, Stabilization of dispersion-generalized Benjamin-Ono, Nonlinear Dispersive Waves and Fluids, Contemp. Math., 725 (2019), 111-136.  doi: 10.1090/conm/725/14548.

[13]

C. Heil, A Basis Theory Primer, Expanded Edition, Applied and Numerical Harmonic Analysis, Birkhauser, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-4687-5.

[14]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Com. Part. Dif. Eq., 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.

[15]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.

[16]

R. J. Jr. Iorio, KdV, BO and friends in weighted Sobolev spaces, Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989), Lecture Notes in Math., Springer, Berlin, 1450 (1990), 104–121. doi: 10.1007/BFb0084901.

[17]

R. J. Jr. Iorio and V. Magalhães, Fourier Analysis and Partial Differential Equations, Cambrige Universiy Press 2001. doi: 10.1017/CBO9780511623745.

[18]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. 

[19]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[20]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics 2005.

[21]

C. Laurent, Internal control of the Schrödinger equation, Math. Control Relat. Fields, 4 (2014), 161-186.  doi: 10.3934/mcrf.2014.4.161.

[22]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM Control Optm. Cal. Var., 16 (2010), 356-379.  doi: 10.1051/cocv/2009001.

[23]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in $L^{2}(\mathbb{T})$, Arch. Rational Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

[24]

C. LaurentL. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[25]

F. Linares and J. H. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM: Cont. Optm. Cal. Var., 11 (2005), 204-218.  doi: 10.1051/cocv:2005002.

[26]

F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain, Trans. Amer. Math. Soc., 367 (2015), 4595-4626.  doi: 10.1090/S0002-9947-2015-06086-3.

[27]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.

[28]

G. P. MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[29]

S. MicuJ. OrtegaL. Rosier and B-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.

[30]

P. D. Miller, P. A. Perry, J.-C. Saut and C. Sulem, Nonlinear Dispersive Partial Differential Equations and Inverse scattering, Fields Institute Comm. 83, Springer, 2019.

[31]

M. Panthee and F. Vielma Leal, On the controllability and stabilization of the linearized Benjamin equation on a periodic domain, Nonlinear Anal. Real World Appl., 51 (2020), 102978.  doi: 10.1016/j.nonrwa.2019.102978.

[32]

M. Panthee and F. Vielma Leal, On the controllability and stabilization of the Benjamin equation on a periodic domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 1605-1652.  doi: 10.1016/j.anihpc.2020.12.004.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[34]

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.

[35]

L. Rosier and B.-Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schr$\ddot{o}$dinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.  doi: 10.1137/070709578.

[36]

L. Rosier and B.-Y. Zhang, Control and stabilization of the nonlinear Schr$\ddot{o}$dinger equation on rectangles, Math. Models Methods Appl. Sci., 20 (2010), 2293-2347.  doi: 10.1142/S0218202510004933.

[37]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation, SIAM J. Control Optim., 45 (2006), 927-956.  doi: 10.1137/050631409.

[38]

W. Rudin, Functional Analysis, 2$^nd$ edition, McGraw-Hill, Inc., New York, 1991.

[39]

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.

[40]

D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the thrid-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.  doi: 10.1137/0331030.

[41]

D. L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[42]

V. I. Shrira and V. V. Voronovich, Nonlinear dynamics of vorticity waves in the coastal zone, J. Fluid Mech., 326 (1996), 181-203.  doi: 10.1017/S0022112096008282.

[43]

M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), 500-508.  doi: 10.1137/0312038.

[44]

R. Smith, Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 57 (1972), 379-391. 

[45]

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]

L. AbdelouhabJ. L. BonaM. Fell and J-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D., 40 (1989), 360-392.  doi: 10.1016/0167-2789(89)90050-X.

[2]

A. AnkiewiczD. J. KedzioraA. ChowduryU. Bandelow and N. Akhmediev, Infinite hierarchy of nonlinear Schrödinger equations and their solutions, Phys. Rev. E., 93 (2016), 012206.  doi: 10.1103/physreve.93.012206.

[3]

J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587.  doi: 10.1002/cpa.3160320405.

[4]

R. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic Schrödinger equation, Appl. Math. Optim., 84 (2021), 103-144. 

[5]

R. Capistrano-Filho and A. Gomes, Global control aspects for long waves in nonlinear dispersive media, preprint, arXiv: 2013.00921v1.

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolutions Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.

[7]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, Amer. Math. Soc., 2007. doi: 10.1090/surv/136.

[8]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. 

[9]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schr$\ddot{o}$dinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.

[10]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[11]

C. Flores, Control and stability of the linearized dispersion-generalized Benjamin-Ono equation on a periodic domain, Math. Control Signals Systems, 30 (2018), Art. 13, 16pp. doi: 10.1007/s00498-018-0219-z.

[12]

C. FloresS. Oh and D. Smith, Stabilization of dispersion-generalized Benjamin-Ono, Nonlinear Dispersive Waves and Fluids, Contemp. Math., 725 (2019), 111-136.  doi: 10.1090/conm/725/14548.

[13]

C. Heil, A Basis Theory Primer, Expanded Edition, Applied and Numerical Harmonic Analysis, Birkhauser, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-4687-5.

[14]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Com. Part. Dif. Eq., 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.

[15]

A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.

[16]

R. J. Jr. Iorio, KdV, BO and friends in weighted Sobolev spaces, Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989), Lecture Notes in Math., Springer, Berlin, 1450 (1990), 104–121. doi: 10.1007/BFb0084901.

[17]

R. J. Jr. Iorio and V. Magalhães, Fourier Analysis and Partial Differential Equations, Cambrige Universiy Press 2001. doi: 10.1017/CBO9780511623745.

[18]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. 

[19]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[20]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics 2005.

[21]

C. Laurent, Internal control of the Schrödinger equation, Math. Control Relat. Fields, 4 (2014), 161-186.  doi: 10.3934/mcrf.2014.4.161.

[22]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM Control Optm. Cal. Var., 16 (2010), 356-379.  doi: 10.1051/cocv/2009001.

[23]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in $L^{2}(\mathbb{T})$, Arch. Rational Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

[24]

C. LaurentL. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[25]

F. Linares and J. H. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM: Cont. Optm. Cal. Var., 11 (2005), 204-218.  doi: 10.1051/cocv:2005002.

[26]

F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain, Trans. Amer. Math. Soc., 367 (2015), 4595-4626.  doi: 10.1090/S0002-9947-2015-06086-3.

[27]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.

[28]

G. P. MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[29]

S. MicuJ. OrtegaL. Rosier and B-Y. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.

[30]

P. D. Miller, P. A. Perry, J.-C. Saut and C. Sulem, Nonlinear Dispersive Partial Differential Equations and Inverse scattering, Fields Institute Comm. 83, Springer, 2019.

[31]

M. Panthee and F. Vielma Leal, On the controllability and stabilization of the linearized Benjamin equation on a periodic domain, Nonlinear Anal. Real World Appl., 51 (2020), 102978.  doi: 10.1016/j.nonrwa.2019.102978.

[32]

M. Panthee and F. Vielma Leal, On the controllability and stabilization of the Benjamin equation on a periodic domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 1605-1652.  doi: 10.1016/j.anihpc.2020.12.004.

[33]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[34]

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.

[35]

L. Rosier and B.-Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schr$\ddot{o}$dinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.  doi: 10.1137/070709578.

[36]

L. Rosier and B.-Y. Zhang, Control and stabilization of the nonlinear Schr$\ddot{o}$dinger equation on rectangles, Math. Models Methods Appl. Sci., 20 (2010), 2293-2347.  doi: 10.1142/S0218202510004933.

[37]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation, SIAM J. Control Optim., 45 (2006), 927-956.  doi: 10.1137/050631409.

[38]

W. Rudin, Functional Analysis, 2$^nd$ edition, McGraw-Hill, Inc., New York, 1991.

[39]

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.

[40]

D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the thrid-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.  doi: 10.1137/0331030.

[41]

D. L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[42]

V. I. Shrira and V. V. Voronovich, Nonlinear dynamics of vorticity waves in the coastal zone, J. Fluid Mech., 326 (1996), 181-203.  doi: 10.1017/S0022112096008282.

[43]

M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), 500-508.  doi: 10.1137/0312038.

[44]

R. Smith, Nonlinear Kelvin and continental-shelf waves, J. Fluid Mech., 57 (1972), 379-391. 

[45]

B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim., 37 (1999), 543-565.  doi: 10.1137/S0363012997327501.

Figure 1.  Dispersion of $ \lambda_{k} $'s for KdV, Benjamin-Ono and Benjamin equations
Figure 2.  Dispersion of $ \lambda_{k} $'s for Schrödinger equation
Figure 3.  Dispersion of $ \lambda_{k} $'s for the Smith and fourth-order Schrödinger equations with $ \mu>0. $
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