# American Institute of Mathematical Sciences

September  2018, 7(3): 403-415. doi: 10.3934/eect.2018020

## Exact boundary controllability for the Boussinesq equation with variable coefficients

 Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia, Mathematical Engineering Laboratory, (LR01ES13), Tunisia Polytechnic School, Tunisia

Received  October 2017 Revised  April 2018 Published  July 2018

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters:
 \left\{ \begin{align} & \rho (x){{y}_{tt}}=-{{(\sigma (x){{y}_{xx}})}_{xx}}+{{(q(x){{y}_{x}})}_{x}}-{{({{y}^{2}})}_{xx}},\ \ \ \ \ \ \ t>0,~x\in (0,l), \\ & y(t,0)={{y}_{xx}}(t,0)=y(t,l)=0,~~\sigma (l){{y}_{xx}}(t,l)=u(t)\ \ \ \ \ t>0, \\ \end{align} \right.
where
 $l>0$
, the coefficients
 $ρ(x)>0, \sigma (x)>0$
,
 $q(x)≥0$
in
 $\left[ {0,l} \right]$
and
 $u$
is the control acting at the end
 $x=l$
. We prove that the linearized problem is exactly controllable in any time
 $T>0$
. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument. This problem has been studied by Crépeau [Diff. Integ. Equat., 2002] in the case of constant coefficients
 $ρ\equiv\sigma \equiv q\equiv1$
.
Citation: Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
##### References:
 [1] D. Banks and G. Kurowski, A Prüfer transformation for the equation of a vibrating beam, Trans. Amer. Math. Soc., 199 (1974), 203-222. doi: 10.2307/1996883. Google Scholar [2] D. Banks and G. Kurowski, A Prüfer transformation for the equation of a vibrating beam subject to axial forces, Journal of Diff. Equat., 24 (1977), 57-74. doi: 10.1016/0022-0396(77)90170-X. Google Scholar [3] J. Ben Amara and A. A. Vladimirov, On a fourth-order problem with spectral and physical parameters in the boundary condition, Izvestiya: Mathematics, 68 (2004), 645-658. doi: 10.1070/IM2004v068n04ABEH000494. Google Scholar [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. Google Scholar [5] J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108. Google Scholar [6] E. Cerpa and E. Crépeau, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, preprint.Google Scholar [7] J. M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Soc., Providence, RI, 2007. Google Scholar [8] E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Diff. Integ. Equat., 16 (2002), 303-326. Google Scholar [9] P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628. doi: 10.1002/cpa.3160350502. Google Scholar [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1983. doi: 10. 1007/978-3-642-58016-1. Google Scholar [11] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures et Appl., 68 (1989), 457-465. Google Scholar [12] A. A. Himonas and D. Mantzavinos, The "good" Boussinesq equation on the half-line, Journal of Diff. Equat., 258 (2015), 3107-3160. doi: 10.1016/j.jde.2015.01.005. Google Scholar [13] N. Kishimoto, Sharp local well-posedness for the "good" Boussinesq equation, Journal of Diff. Equat., 254 (2013), 2393-2433. doi: 10.1016/j.jde.2012.12.008. Google Scholar [14] V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, John Wiley-Masson, 1994. Google Scholar [15] W. Leighton and Z. Nehari, On the oscillation of solutions of self-adjoint linear differential equations of fourth-order, Trans. Amer. Math. Soc., 89 (1958), 325-377. doi: 10.1090/S0002-9947-1958-0102639-X. Google Scholar [16] B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory, AMS, 1975. Google Scholar [17] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, Journal of Diff. Equat., 106 (1993), 257-293. doi: 10.1006/jdeq.1993.1108. Google Scholar [18] J. L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. Google Scholar [19] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. Google Scholar [20] J. L. Lions and E. Magenes, Non–Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972. Google Scholar [21] M. A. Naimark, Linear Differential Operators, Ungar, New York, 167.Google Scholar [22] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: COCV, 2 (1997), 33-55. doi: 10.1051/cocv:1997102. Google Scholar [23] S. K. Turitsyn, Nonstable solitons and sharp criteria for wave collapse, Phys. Rev. E, 47 (1993), R13-R16. doi: 10.1103/PhysRevE.47.R13. Google Scholar [24] V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP., 38 (1974), 108-110. Google Scholar [25] B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, in Control and Estimation of Distributed Parameter Systems, 126 (1988), 297–310. Google Scholar

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##### References:
 [1] D. Banks and G. Kurowski, A Prüfer transformation for the equation of a vibrating beam, Trans. Amer. Math. Soc., 199 (1974), 203-222. doi: 10.2307/1996883. Google Scholar [2] D. Banks and G. Kurowski, A Prüfer transformation for the equation of a vibrating beam subject to axial forces, Journal of Diff. Equat., 24 (1977), 57-74. doi: 10.1016/0022-0396(77)90170-X. Google Scholar [3] J. Ben Amara and A. A. Vladimirov, On a fourth-order problem with spectral and physical parameters in the boundary condition, Izvestiya: Mathematics, 68 (2004), 645-658. doi: 10.1070/IM2004v068n04ABEH000494. Google Scholar [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. Google Scholar [5] J. V. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17 (1872), 55-108. Google Scholar [6] E. Cerpa and E. Crépeau, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, preprint.Google Scholar [7] J. M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Soc., Providence, RI, 2007. Google Scholar [8] E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Diff. Integ. Equat., 16 (2002), 303-326. Google Scholar [9] P. Deift, C. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628. doi: 10.1002/cpa.3160350502. Google Scholar [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1983. doi: 10. 1007/978-3-642-58016-1. Google Scholar [11] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures et Appl., 68 (1989), 457-465. Google Scholar [12] A. A. Himonas and D. Mantzavinos, The "good" Boussinesq equation on the half-line, Journal of Diff. Equat., 258 (2015), 3107-3160. doi: 10.1016/j.jde.2015.01.005. Google Scholar [13] N. Kishimoto, Sharp local well-posedness for the "good" Boussinesq equation, Journal of Diff. Equat., 254 (2013), 2393-2433. doi: 10.1016/j.jde.2012.12.008. Google Scholar [14] V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, John Wiley-Masson, 1994. Google Scholar [15] W. Leighton and Z. Nehari, On the oscillation of solutions of self-adjoint linear differential equations of fourth-order, Trans. Amer. Math. Soc., 89 (1958), 325-377. doi: 10.1090/S0002-9947-1958-0102639-X. Google Scholar [16] B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory, AMS, 1975. Google Scholar [17] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, Journal of Diff. Equat., 106 (1993), 257-293. doi: 10.1006/jdeq.1993.1108. Google Scholar [18] J. L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. Google Scholar [19] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. Google Scholar [20] J. L. Lions and E. Magenes, Non–Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972. Google Scholar [21] M. A. Naimark, Linear Differential Operators, Ungar, New York, 167.Google Scholar [22] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: COCV, 2 (1997), 33-55. doi: 10.1051/cocv:1997102. Google Scholar [23] S. K. Turitsyn, Nonstable solitons and sharp criteria for wave collapse, Phys. Rev. E, 47 (1993), R13-R16. doi: 10.1103/PhysRevE.47.R13. Google Scholar [24] V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP., 38 (1974), 108-110. Google Scholar [25] B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, in Control and Estimation of Distributed Parameter Systems, 126 (1988), 297–310. Google Scholar
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