June  2022, 15(6): 1499-1523. doi: 10.3934/dcdss.2022096

On the global controllability of the 1-D Boussinesq equation

1. 

Ecole Nationale d'Ingénieurs de Tunis, Université de Tunis El-Manar & Laboratoire d'Ingénierie, Mathématique (LIM), Ecole Polytechnique de Tunisie, Université de Carthage, Tunisia

2. 

Faculté des Sciences de Bizerte, Université de Carthage & UR Analysis and Control of PDEs, UR 13ES64, University of Monastir, Tunisia

* Corresponding author: Chaker Jammazi

The first author is supported by LIM Laboratory of EPT

Received  August 2021 Revised  March 2022 Published  June 2022 Early access  April 2022

We prove in this paper the global approximate controllability of the 1-D Boussinesq equation-subjected to internal control and free boundary conditions-on a bounded domain. The key ingredients of the proof relies Coron's return method for the exact global controllability of the nonlinear control system $ y_{tt}+(y^2)_{xx} = u(t) $, combined with some priori estimates for nonlinear weak-hyperbolic systems defined respectively in Gevrey class of functions, and in Sobolev spaces.

Citation: Chaker Jammazi, Souhila Loucif. On the global controllability of the 1-D Boussinesq equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1499-1523. doi: 10.3934/dcdss.2022096
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

A. Arosio and S. Spagnolo, Global existance for abstract evolution equations of weakly hyperbolic type, J. Math. Pures Appl., 65 (1986), 263-305. 

[3]

K. Beauchard, Contribution à L'étude de la Contrôlabilité et de la Stabilisation de L'équation de Schrödinger, PhD thesis, Université d'Orsay, 2005.

[4]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.

[5]

E. Cerpa and E. Crépeau, On the controllability of the improved Boussinesq equation, SIAM J. Control Optim., 56 (2018), 3035-3049.  doi: 10.1137/16M108923X.

[6]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, Journal of Evolution Equations, 18 (2018), 1501-1519.  doi: 10.1007/s00028-018-0450-6.

[7]

M. Chapouly, Contrôlabilité D'équations Issues de la Mécanique des Fluides, Thèse de doctorat, Université Paris-Sud-Orsay, 2009.

[8]

M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim., 48 (2009), 1567-1599.  doi: 10.1137/070685749.

[9]

H. R. Clark, Global classical solutions to the Cauchy problem for a nonlinear wave equation, Internat. J. Math. Math. Sci., 21 (1998), 533-548.  doi: 10.1155/S016117129800074X.

[10]

F. ColombiniE. De Giorgio and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Della Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. 

[11]

F. ColombiniD. Del Santo and T. Kinoshita, Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients, J. Math. Pures Appl., 81 (2002), 641-654.  doi: 10.1016/S0021-7824(01)01252-1.

[12]

F. ColombiniE. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time, Ann. Della Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 291-312. 

[13]

F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in ${C^{\infty}}$, Acta Math., 148 (1982), 243-253.  doi: 10.1007/BF02392730.

[14]

F. Colombini and S. Spagnolo, Some examples of hyperbolic equation without local solvability, Ann. Sci. École Norm. Sup., 22 (1989), 109-125.  doi: 10.24033/asens.1578.

[15]

J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C.R. Acad. Sci. Paris, 317 (1993), 271-276. 

[16]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[17]

J.-M. Coron, On the controllability of nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, India, World Scientific Publishing Co Pte Ltd, 2010,Vol. Ⅰ: Plenary Lectures and Ceremonies, 1 (2010), 238-264. 

[18]

J.-M. CoronJ. I. DiazA. Drici and T. Mingazzin, Global null controllability of the 1-dimensionnal nonlinear slow diffusion equation, Chinese Ann. Math. Ser. B, 34 (2013), 333-344.  doi: 10.1007/s11401-013-0774-z.

[19]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[20]

E. Crépeau, Contrôlabilité Exacte D'équations Dispersives Issues de la Mécaniques, Thèse de Doctorat, Université Paris-Sud-Orsay, 2002.

[21]

E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Differential and Integral Equations, 16 (2003), 303-326. 

[22]

P. D'Ancona, Private communication.,

[23]

P. D'Ancona and M. Reissig, New trends in the theory of nonlinear weakly hyperbolic equation of second order, Nonlinear Anal., 30 (1997), 2507-2515.  doi: 10.1016/S0362-546X(96)00354-9.

[24]

W. DörflerH. Gerner and R. Schnaubelt, Local well-posedness of a quasilinear wave equation, Appl. Anal., 95 (2016), 2110-2123.  doi: 10.1080/00036811.2015.1089236.

[25]

P. Gao, Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 9 (2020), 181-191.  doi: 10.3934/eect.2020002.

[26]

O. Glass, On the controllability of the Vlasov-Poisson system, J. Differential Equations, 195 (2003), 332-379.  doi: 10.1016/S0022-0396(03)00066-4.

[27]

H. Hermes, Large-time local controllability via homogeneous approximations, SIAM J. Contol And Optimization, 34 (1996), 1291-1299.  doi: 10.1137/S0363012994268059.

[28]

F. John, Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure and Appl. Math., 29 (1976), 649-682.  doi: 10.1002/cpa.3160290608.

[29]

K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J., 12 (1983), 434-460.  doi: 10.14492/hokmj/1525852966.

[30]

M. Kawski, High-order small-time local controllability, Nonlinear Controllability and Optimal Control, Marcel Dekker, Monogr. Textbooks Pure Appl. Math., Dekker, New York, 133 (1990), 431-467. 

[31]

S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure and Appl. Math., 33 (1980), 43-101.  doi: 10.1002/cpa.3160330104.

[32]

H. LiQ. Lü and X. Zhang, Recent progress on controllability / observability for systems governed by partial differential equations, L. Syst. Sci. Complex, 23 (2010), 527-545.  doi: 10.1007/s11424-010-0144-9.

[33]

S. LiM. Chen and B. Zhang, Controllability and stabilizability of a higher order wave equation on a periodic domain, SIAM J. Control Optim., 58 (2020), 1121-1143.  doi: 10.1137/19M1240472.

[34]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[35]

M. Reissig and K. Yagdjian, Levi conditions and global Gevrey regularity for the solutions of quasilinear weakly hyperbolic equations, Math. Nachr., 178 (1996), 285-307.  doi: 10.1002/mana.19961780114.

[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]

D. L. Russell and Y. B. 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.

[38]

J. WuX. Zhu and S. Chai, Controllability for one-dimensional nonlinear wave equations with degenarte damping, Systems and Control Letters, 100 (2017), 66-72.  doi: 10.1016/j.sysconle.2016.12.007.

[39]

B.-Y. Zhang, Exact controllability of the generalized Boussinesq equation, Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 126 (1988), 297-311. 

[40]

X. Zhang, Remarks on the controllability of some quasilinear equations, Some Problems on Nonlinear Hyperbolic Equations and Applications, Ser. Contemp. Appl. Math. CAM, Higher Ed. Press, Beijing, 15 (2010), 437-452.  doi: 10.1142/9789814322898_0020.

[41]

Y. Zhou and Z. Lei, Local exact boundary controllability for nonlinear wave equations, SIAM J. Control. Optim., 46 (2007), 1022-1051.  doi: 10.1137/060650222.

[42]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/s0294-1449(16)30221-9.

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

A. Arosio and S. Spagnolo, Global existance for abstract evolution equations of weakly hyperbolic type, J. Math. Pures Appl., 65 (1986), 263-305. 

[3]

K. Beauchard, Contribution à L'étude de la Contrôlabilité et de la Stabilisation de L'équation de Schrödinger, PhD thesis, Université d'Orsay, 2005.

[4]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Commun. Math. Phys., 118 (1988), 15-29.  doi: 10.1007/BF01218475.

[5]

E. Cerpa and E. Crépeau, On the controllability of the improved Boussinesq equation, SIAM J. Control Optim., 56 (2018), 3035-3049.  doi: 10.1137/16M108923X.

[6]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, Journal of Evolution Equations, 18 (2018), 1501-1519.  doi: 10.1007/s00028-018-0450-6.

[7]

M. Chapouly, Contrôlabilité D'équations Issues de la Mécanique des Fluides, Thèse de doctorat, Université Paris-Sud-Orsay, 2009.

[8]

M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim., 48 (2009), 1567-1599.  doi: 10.1137/070685749.

[9]

H. R. Clark, Global classical solutions to the Cauchy problem for a nonlinear wave equation, Internat. J. Math. Math. Sci., 21 (1998), 533-548.  doi: 10.1155/S016117129800074X.

[10]

F. ColombiniE. De Giorgio and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Della Scuola Norm. Sup. Pisa Cl. Sci., 6 (1979), 511-559. 

[11]

F. ColombiniD. Del Santo and T. Kinoshita, Gevrey-well-posedness for weakly hyperbolic operators with non-regular coefficients, J. Math. Pures Appl., 81 (2002), 641-654.  doi: 10.1016/S0021-7824(01)01252-1.

[12]

F. ColombiniE. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time, Ann. Della Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 291-312. 

[13]

F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in ${C^{\infty}}$, Acta Math., 148 (1982), 243-253.  doi: 10.1007/BF02392730.

[14]

F. Colombini and S. Spagnolo, Some examples of hyperbolic equation without local solvability, Ann. Sci. École Norm. Sup., 22 (1989), 109-125.  doi: 10.24033/asens.1578.

[15]

J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C.R. Acad. Sci. Paris, 317 (1993), 271-276. 

[16]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[17]

J.-M. Coron, On the controllability of nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, India, World Scientific Publishing Co Pte Ltd, 2010,Vol. Ⅰ: Plenary Lectures and Ceremonies, 1 (2010), 238-264. 

[18]

J.-M. CoronJ. I. DiazA. Drici and T. Mingazzin, Global null controllability of the 1-dimensionnal nonlinear slow diffusion equation, Chinese Ann. Math. Ser. B, 34 (2013), 333-344.  doi: 10.1007/s11401-013-0774-z.

[19]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[20]

E. Crépeau, Contrôlabilité Exacte D'équations Dispersives Issues de la Mécaniques, Thèse de Doctorat, Université Paris-Sud-Orsay, 2002.

[21]

E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Differential and Integral Equations, 16 (2003), 303-326. 

[22]

P. D'Ancona, Private communication.,

[23]

P. D'Ancona and M. Reissig, New trends in the theory of nonlinear weakly hyperbolic equation of second order, Nonlinear Anal., 30 (1997), 2507-2515.  doi: 10.1016/S0362-546X(96)00354-9.

[24]

W. DörflerH. Gerner and R. Schnaubelt, Local well-posedness of a quasilinear wave equation, Appl. Anal., 95 (2016), 2110-2123.  doi: 10.1080/00036811.2015.1089236.

[25]

P. Gao, Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 9 (2020), 181-191.  doi: 10.3934/eect.2020002.

[26]

O. Glass, On the controllability of the Vlasov-Poisson system, J. Differential Equations, 195 (2003), 332-379.  doi: 10.1016/S0022-0396(03)00066-4.

[27]

H. Hermes, Large-time local controllability via homogeneous approximations, SIAM J. Contol And Optimization, 34 (1996), 1291-1299.  doi: 10.1137/S0363012994268059.

[28]

F. John, Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions, Comm. Pure and Appl. Math., 29 (1976), 649-682.  doi: 10.1002/cpa.3160290608.

[29]

K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J., 12 (1983), 434-460.  doi: 10.14492/hokmj/1525852966.

[30]

M. Kawski, High-order small-time local controllability, Nonlinear Controllability and Optimal Control, Marcel Dekker, Monogr. Textbooks Pure Appl. Math., Dekker, New York, 133 (1990), 431-467. 

[31]

S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure and Appl. Math., 33 (1980), 43-101.  doi: 10.1002/cpa.3160330104.

[32]

H. LiQ. Lü and X. Zhang, Recent progress on controllability / observability for systems governed by partial differential equations, L. Syst. Sci. Complex, 23 (2010), 527-545.  doi: 10.1007/s11424-010-0144-9.

[33]

S. LiM. Chen and B. Zhang, Controllability and stabilizability of a higher order wave equation on a periodic domain, SIAM J. Control Optim., 58 (2020), 1121-1143.  doi: 10.1137/19M1240472.

[34]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[35]

M. Reissig and K. Yagdjian, Levi conditions and global Gevrey regularity for the solutions of quasilinear weakly hyperbolic equations, Math. Nachr., 178 (1996), 285-307.  doi: 10.1002/mana.19961780114.

[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]

D. L. Russell and Y. B. 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.

[38]

J. WuX. Zhu and S. Chai, Controllability for one-dimensional nonlinear wave equations with degenarte damping, Systems and Control Letters, 100 (2017), 66-72.  doi: 10.1016/j.sysconle.2016.12.007.

[39]

B.-Y. Zhang, Exact controllability of the generalized Boussinesq equation, Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 126 (1988), 297-311. 

[40]

X. Zhang, Remarks on the controllability of some quasilinear equations, Some Problems on Nonlinear Hyperbolic Equations and Applications, Ser. Contemp. Appl. Math. CAM, Higher Ed. Press, Beijing, 15 (2010), 437-452.  doi: 10.1142/9789814322898_0020.

[41]

Y. Zhou and Z. Lei, Local exact boundary controllability for nonlinear wave equations, SIAM J. Control. Optim., 46 (2007), 1022-1051.  doi: 10.1137/060650222.

[42]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/s0294-1449(16)30221-9.

Figure 1.  Example of trajectory $ a(t) $
Figure 2.  Local controllability of the system $ y_{tt}+(y^2)_{xx} = u $
Figure 3.  Approximate controllability of Boussinesq type equation
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