Figure 1.
Example of trajectory $ a(t) $
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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 |
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.
Keywords:
Boussinesq equation,
global controllability,
return method,
Gevrey regularity,
approximate controllability,
internal control.
Mathematics Subject Classification:
Primary: 93B05, 93C20, 93C10; Secondary: 35Q35.
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. Colombini, E. 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. Colombini, D. 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. Colombini, E. 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. Coron, J. I. Diaz, A. 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. Dörfler, H. 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. Li, Q. 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. Li, M. 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. Wu, X. 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. Colombini, E. 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. Colombini, D. 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. Colombini, E. 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. Coron, J. I. Diaz, A. 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. Dörfler, H. 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. Li, Q. 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. Li, M. 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. Wu, X. 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 2.
Local controllability of the system $ y_{tt}+(y^2)_{xx} = u $
Figure 3.
Approximate controllability of Boussinesq type equation
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