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  December 2019 Early access  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 and 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.

[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.

[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[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.

[5]

C. BardosG. 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.

[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.

[7]

K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273.  doi: 10.1137/050642034.

[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.

[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.

[10]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[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.

[12]

P. CannarsaG. 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.

[13]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978.

[14]

J. Davis Philip, Interpolation and Approximation, Dover publications, INC., New York, 1975.

[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.

[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.

[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.

[18]

G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, arXiv: 1710.00690.

[19]

X. FuJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Systems, Control and Communications, 1 (2008), 1-12. 

[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.

[29]

P. LinZ. 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.

[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.

[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.

[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.

[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.

[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.

[35]

M. A. ShubovC. F. MartinJ. 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.

[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. 

[37]

J. WuX. 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.

[38]

J. WuX. 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.

[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.

[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.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York San Francisco, London, 1975.

[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.

[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[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.

[5]

C. BardosG. 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.

[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.

[7]

K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273.  doi: 10.1137/050642034.

[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.

[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.

[10]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[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.

[12]

P. CannarsaG. 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.

[13]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, 1978.

[14]

J. Davis Philip, Interpolation and Approximation, Dover publications, INC., New York, 1975.

[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.

[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.

[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.

[18]

G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, arXiv: 1710.00690.

[19]

X. FuJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Systems, Control and Communications, 1 (2008), 1-12. 

[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.

[29]

P. LinZ. 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.

[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.

[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.

[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.

[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.

[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.

[35]

M. A. ShubovC. F. MartinJ. 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.

[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. 

[37]

J. WuX. 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.

[38]

J. WuX. 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.

[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.

[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.

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