# American Institute of Mathematical Sciences

December  2019, 8(4): 669-686. doi: 10.3934/eect.2019039

## Controllability of the semilinear wave equation governed by a multiplicative control

 MASI Laboratory, Department of Mathematics & Informatics, ENS. University of Sidi Mohamed Ben Abdellah, Fes, P.O. Box 5206, Morocco

* Corresponding author: M. Ouzahra

Dedicated to Professor Hammadi Bouslous on the occasion of his 65th birthday

Received  August 2017 Revised  February 2019 Published  June 2019

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

Citation: Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York San Francisco, London, 1975. Google Scholar [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces Applications to PDES and Optimization, SIAM, Society for Industrial and Applied Mathematics, USA, 2006. Google Scholar [3] J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597. doi: 10.1137/0320042. Google Scholar [4] J. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265. doi: 10.1016/0022-0396(78)90032-3. Google Scholar [5] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. Google Scholar [6] K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021. Google Scholar [7] K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273. doi: 10.1137/050642034. Google Scholar [8] K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schr$\ddot{o}$dinger equations with bilinear control, Journal de Mathéematiques Pures et Appliquées, 94 (2010), 520-554. doi: 10.1016/j.matpur.2010.04.001. Google Scholar [9] K. Beauchard, Local controllability and non-controllability for a $1D$ wave equation with bilinear control, J. Differential Equations, 250 (2011), 2064-2098. doi: 10.1016/j.jde.2010.10.008. Google Scholar [10] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. Google Scholar [11] P. Cannarsa and A. Khapalov, Multiplicative controllability for reaction-diffusion equation with target states admitting finitely many changes of sign, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1293-1311. doi: 10.3934/dcdsb.2010.14.1293. Google Scholar [12] P. Cannarsa, G. Floridia and A. Khapalov, Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign, Journal de Mathématiques Pures et Appliquées, 108 (2017), 425-458. doi: 10.1016/j.matpur.2017.07.002. Google Scholar [13] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978. Google Scholar [14] J. Davis Philip, Interpolation and Approximation, Dover publications, INC., New York, 1975. Google Scholar [15] I. El Harraki and A. Boutoulout, Controllability of the wave equation via multiplicative controls, IMA Journal of Mathematical Control and Information, 35 (2016), 393-409. doi: 10.1093/imamci/dnw055. Google Scholar [16] L. A. Fernández and A. Y. Khapalov, Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support, ESAIM: Control Optim. Calc. Var., 18 (2012), 1207-1224. doi: 10.1051/cocv/2012004. Google Scholar [17] G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422. doi: 10.1016/j.jde.2014.06.016. Google Scholar [18] G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, arXiv: 1710.00690.Google Scholar [19] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM Journal on Control and Optimization, 46 (2007), 1578-1614. doi: 10.1137/040610222. Google Scholar [20] A. Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control. Optim., 41 (2003), 1886-1900. doi: 10.1137/S0363012901394607. Google Scholar [21] A. Y. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Cont. Dyn. Syst., 11 (2004), 311-324. doi: 10.3934/dcds.2004.11.311. Google Scholar [22] A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM: Control Optim. Calc. Var., 12 (2006), 231-252. doi: 10.1051/cocv:2006001. Google Scholar [23] A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12413-6. Google Scholar [24] K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems and Control Letters., 24 (1995), 301-306. doi: 10.1016/0167-6911(94)00022-N. Google Scholar [25] V. Komornik, Exact controllability in short time for the wave equation, Annales de l'Institut Henri Poincaré, 6 (1989), 153-164. doi: 10.1016/S0294-1449(16)30327-4. Google Scholar [26] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization, 23 (1991), 109-154. doi: 10.1007/BF01442394. Google Scholar [27] H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Systems, Control and Communications, 1 (2008), 1-12. Google Scholar [28] M. Liang, Bilinear optimal control for a wave equation, Mathematical Models and Methods in Applied Sciences, 9 (1999), 45-68. doi: 10.1142/S0218202599000051. Google Scholar [29] P. Lin, Z. Zhou and H. Gao, Exact controllability of the parabolic system with bilinear control, Applied Mathematics Letters, 19 (2006), 568-575. doi: 10.1016/j.aml.2005.05.016. Google Scholar [30] P. Martinez and J. Vancostenoble, Exact controllability in "arbitrarily short time" of the semilinear wave equation, Discrete and Continuous Dynamical Systems, 9 (2003), 901-924. doi: 10.3934/dcds.2003.9.901. Google Scholar [31] V. Nersesyan, Growth of Sobolev norms and controllability of the Schr$\ddot{o}$dinger equation, Comm. Math. Phys., 290 (2009), 371-387. doi: 10.1007/s00220-009-0842-0. Google Scholar [32] M. Ouzahra, Controllability of the wave equation with bilinear controls, European Journal of Control, 20 (2014), 57-63. doi: 10.1016/j.ejcon.2013.10.007. Google Scholar [33] M. Ouzahra, Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control, European Journal of Control, 32 (2016), 32-38. doi: 10.1016/j.ejcon.2016.05.004. Google Scholar [34] A. Pazy, Semi-groups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [35] M. A. Shubov, C. F. Martin, J. P. Dauer and B. P. Belinskiy, Exact controllability of the damped wave equation, SIAM J. Control Optim., 35 (1997), 1773-1789. doi: 10.1137/S0363012996291616. Google Scholar [36] L. Tebou, Equivalence between observability and stabilization for a class of second order semilinear evolution equation, Discrete and Continuous Dynamical Systems, 1 (2009), 744-752. Google Scholar [37] J. Wu, X. Zhu and S. Li, Simultaneous controllability of damped wave equations, Mathematical Methods in the Applied Sciences, 40 (2017), 319-324. doi: 10.1002/mma.4175. Google Scholar [38] J. Wu, X. Zhu and S. Chai, Controllability for one-dimensional nonlinear wave equations with degenerate damping, Systems and Control Letters, 100 (2017), 66-72. doi: 10.1016/j.sysconle.2016.12.007. Google Scholar [39] X. Zhang, Exact controllability of semilinear evolution systems and its application, Journal of Optimization Theory and Applications, 107 (2000), 415-432. doi: 10.1023/A:1026460831701. Google Scholar [40] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Annales de l'I.H.P, Analyse non linéaire, tome, 10 (1993), 109–129. doi: 10.1016/S0294-1449(16)30221-9. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York San Francisco, London, 1975. Google Scholar [2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces Applications to PDES and Optimization, SIAM, Society for Industrial and Applied Mathematics, USA, 2006. Google Scholar [3] J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597. doi: 10.1137/0320042. Google Scholar [4] J. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265. doi: 10.1016/0022-0396(78)90032-3. Google Scholar [5] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. Google Scholar [6] K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021. Google Scholar [7] K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273. doi: 10.1137/050642034. Google Scholar [8] K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schr$\ddot{o}$dinger equations with bilinear control, Journal de Mathéematiques Pures et Appliquées, 94 (2010), 520-554. doi: 10.1016/j.matpur.2010.04.001. Google Scholar [9] K. Beauchard, Local controllability and non-controllability for a $1D$ wave equation with bilinear control, J. Differential Equations, 250 (2011), 2064-2098. doi: 10.1016/j.jde.2010.10.008. Google Scholar [10] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. Google Scholar [11] P. Cannarsa and A. Khapalov, Multiplicative controllability for reaction-diffusion equation with target states admitting finitely many changes of sign, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1293-1311. doi: 10.3934/dcdsb.2010.14.1293. Google Scholar [12] P. Cannarsa, G. Floridia and A. Khapalov, Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign, Journal de Mathématiques Pures et Appliquées, 108 (2017), 425-458. doi: 10.1016/j.matpur.2017.07.002. Google Scholar [13] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978. Google Scholar [14] J. Davis Philip, Interpolation and Approximation, Dover publications, INC., New York, 1975. Google Scholar [15] I. El Harraki and A. Boutoulout, Controllability of the wave equation via multiplicative controls, IMA Journal of Mathematical Control and Information, 35 (2016), 393-409. doi: 10.1093/imamci/dnw055. Google Scholar [16] L. A. Fernández and A. Y. Khapalov, Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support, ESAIM: Control Optim. Calc. Var., 18 (2012), 1207-1224. doi: 10.1051/cocv/2012004. Google Scholar [17] G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422. doi: 10.1016/j.jde.2014.06.016. Google Scholar [18] G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, arXiv: 1710.00690.Google Scholar [19] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM Journal on Control and Optimization, 46 (2007), 1578-1614. doi: 10.1137/040610222. Google Scholar [20] A. Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control. Optim., 41 (2003), 1886-1900. doi: 10.1137/S0363012901394607. Google Scholar [21] A. Y. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Cont. Dyn. Syst., 11 (2004), 311-324. doi: 10.3934/dcds.2004.11.311. Google Scholar [22] A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM: Control Optim. Calc. Var., 12 (2006), 231-252. doi: 10.1051/cocv:2006001. Google Scholar [23] A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12413-6. Google Scholar [24] K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems and Control Letters., 24 (1995), 301-306. doi: 10.1016/0167-6911(94)00022-N. Google Scholar [25] V. Komornik, Exact controllability in short time for the wave equation, Annales de l'Institut Henri Poincaré, 6 (1989), 153-164. doi: 10.1016/S0294-1449(16)30327-4. Google Scholar [26] I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization, 23 (1991), 109-154. doi: 10.1007/BF01442394. Google Scholar [27] H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Systems, Control and Communications, 1 (2008), 1-12. Google Scholar [28] M. Liang, Bilinear optimal control for a wave equation, Mathematical Models and Methods in Applied Sciences, 9 (1999), 45-68. doi: 10.1142/S0218202599000051. Google Scholar [29] P. Lin, Z. Zhou and H. Gao, Exact controllability of the parabolic system with bilinear control, Applied Mathematics Letters, 19 (2006), 568-575. doi: 10.1016/j.aml.2005.05.016. Google Scholar [30] P. Martinez and J. Vancostenoble, Exact controllability in "arbitrarily short time" of the semilinear wave equation, Discrete and Continuous Dynamical Systems, 9 (2003), 901-924. doi: 10.3934/dcds.2003.9.901. Google Scholar [31] V. Nersesyan, Growth of Sobolev norms and controllability of the Schr$\ddot{o}$dinger equation, Comm. Math. Phys., 290 (2009), 371-387. doi: 10.1007/s00220-009-0842-0. Google Scholar [32] M. Ouzahra, Controllability of the wave equation with bilinear controls, European Journal of Control, 20 (2014), 57-63. doi: 10.1016/j.ejcon.2013.10.007. Google Scholar [33] M. Ouzahra, Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control, European Journal of Control, 32 (2016), 32-38. doi: 10.1016/j.ejcon.2016.05.004. Google Scholar [34] A. Pazy, Semi-groups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [35] M. A. Shubov, C. F. Martin, J. P. Dauer and B. P. Belinskiy, Exact controllability of the damped wave equation, SIAM J. Control Optim., 35 (1997), 1773-1789. doi: 10.1137/S0363012996291616. Google Scholar [36] L. Tebou, Equivalence between observability and stabilization for a class of second order semilinear evolution equation, Discrete and Continuous Dynamical Systems, 1 (2009), 744-752. Google Scholar [37] J. Wu, X. Zhu and S. Li, Simultaneous controllability of damped wave equations, Mathematical Methods in the Applied Sciences, 40 (2017), 319-324. doi: 10.1002/mma.4175. Google Scholar [38] J. Wu, X. Zhu and S. Chai, Controllability for one-dimensional nonlinear wave equations with degenerate damping, Systems and Control Letters, 100 (2017), 66-72. doi: 10.1016/j.sysconle.2016.12.007. Google Scholar [39] X. Zhang, Exact controllability of semilinear evolution systems and its application, Journal of Optimization Theory and Applications, 107 (2000), 415-432. doi: 10.1023/A:1026460831701. Google Scholar [40] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Annales de l'I.H.P, Analyse non linéaire, tome, 10 (1993), 109–129. doi: 10.1016/S0294-1449(16)30221-9. Google Scholar
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