# 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 and 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. [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. [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. [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] 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. [6] E. Cerpa and E. Crépeau, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, preprint. [7] J. M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Soc., Providence, RI, 2007. [8] E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Diff. Integ. Equat., 16 (2002), 303-326. [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. [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1983. doi: 10. 1007/978-3-642-58016-1. [11] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures et Appl., 68 (1989), 457-465. [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. [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. [14] V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, John Wiley-Masson, 1994. [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. [16] B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory, AMS, 1975. [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. [18] J. L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001. [19] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. [20] J. L. Lions and E. Magenes, Non–Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972. [21] M. A. Naimark, Linear Differential Operators, Ungar, New York, 167. [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. [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. [24] V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP., 38 (1974), 108-110. [25] B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, in Control and Estimation of Distributed Parameter Systems, 126 (1988), 297–310.

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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. [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. [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. [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] 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. [6] E. Cerpa and E. Crépeau, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, preprint. [7] J. M. Coron, Control and Nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Soc., Providence, RI, 2007. [8] E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Diff. Integ. Equat., 16 (2002), 303-326. [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. [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, 1983. doi: 10. 1007/978-3-642-58016-1. [11] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures et Appl., 68 (1989), 457-465. [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. [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. [14] V. Komornik, Exact Controllability and Stabilization, the Multiplier Method, John Wiley-Masson, 1994. [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. [16] B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory, AMS, 1975. [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. [18] J. L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001. [19] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. [20] J. L. Lions and E. Magenes, Non–Homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972. [21] M. A. Naimark, Linear Differential Operators, Ungar, New York, 167. [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. [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. [24] V. E. Zakharov, On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP., 38 (1974), 108-110. [25] B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, in Control and Estimation of Distributed Parameter Systems, 126 (1988), 297–310.
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