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doi: 10.3934/dcdsb.2021081

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

Mohamed Ouzahra ,

M2PA Laboratory, ENS, University of Sidi Mohamed Ben Abdellah, P.O. Box 5206, Bensouda, Fès, Morocco

* Corresponding author: Mohamed Ouzahra

Received  April 2020 Revised  August 2020 Early access  March 2021

Fund Project: The author is supported by USMBA grant (2020)

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.

Keywords: Semilinear equation, reaction-diffusion, approximate controllability, bilinear control.
Mathematics Subject Classification: Primary: 35K57, 35K58; Secondary: 93C20.
Citation: Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021081
References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.  Google Scholar

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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, Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.  Google Scholar

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J. M. 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

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J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169-179.  doi: 10.1007/BF01442552.  Google Scholar

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J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society, 63 (1977) 370–373. doi: 10.2307/2041821.  Google Scholar

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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

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H. T. Banks, P. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two-dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.  doi: 10.1093/ee/17.5.815.  Google Scholar

[8]

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

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

[10]

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

[11]

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

[12]

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

[13]

P. Cannarsa, G. Floridia and A. Y. 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

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P. J. Davis, Interpolation and Approximation, Dover Publications, INC. New York, 1975.  Google Scholar

[15]

Z. Ditzian, Derivatives of Bernstein polynomials and smoothness, Proceedings of the American Mathematical Society, 93 (1985), 25-31.  doi: 10.1090/S0002-9939-1985-0766520-7.  Google Scholar

[16]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003.  Google Scholar

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A. Doubova, E. Fernández-cara, Go nzález-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM. J. Control Optim, 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.  Google Scholar

[18]

L. C. Evans, Partial Differential Equation, Graduate Studies in Math, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[19]

C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[20]

E. Fern$\acute{a}$ndez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[21]

L. A. Fern$\acute{a}$ndez and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.  Google Scholar

[22]

L. A. Fern$\acute{a}$ndez 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

[23]

S. C. Ferreira, M. L. Martins and M. J. Vilela, Reaction-diffusion model for the growth of avascular tumor, Phys. Rev. E, 65 (2002), 021907. doi: 10.1103/PhysRevE.65.021907.  Google Scholar

[24]

A. Friedman, PDE problems arising in mathematical biology, Networks and Heterogeneousmedia, 7 (2012), 691-703.  doi: 10.3934/nhm.2012.7.691.  Google Scholar

[25]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes 34, Seoul National University, Korea, 1996.  Google Scholar

[26]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numerica, 3, (1994) 269–378. doi: 10.1017/S0962492900002452.  Google Scholar

[27]

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

[28]

A. 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

[29]

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

[30]

A. Y. Khapalov, Controllability of partial differential equations governed by multiplicative controls, Lecture Notes in Mathematics, 1995, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.  Google Scholar

[31]

K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems & Control Letters, 24 (1995), 301-306.  doi: 10.1016/0167-6911(94)00022-N.  Google Scholar

[32]

G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, Comm. PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[33]

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

[34]

G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Co., New York, 1986.  Google Scholar

[35]

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

[36]

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

[37]

M. Ouzahra, A. Tsouli and A. Boutoulout, Exact controllability of the heat equation with bilinear control, Mathematical Methods in the Applied Sciences, 38 (2015), 5074-5084.  doi: 10.1002/mma.3428.  Google Scholar

[38]

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

[39]

M. Ouzahra, Controllability of the semilinear wave equation governed by a multiplicative control, Evolution Equations & Control Theory, 8 (2019), 669-686.  doi: 10.3934/eect.2019039.  Google Scholar

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[41]

B. Perthame, Some mathematical aspects of tumor growth and therapy, Proceedings of the International Congress of Mathematicians—Seoul, Vol. 1, Kyung Moon Sa, Seoul, 2014,529–545.  Google Scholar

[42]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007) 179–208. doi: 10.1137/S0036144504446291.  Google Scholar

[43]

S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag Italia, Milan, 2008.  Google Scholar

[44]

A. Tosin, Initial/boundary-value problems of tumor growth within a host tissue, J. Math.Biol., 66 (2013) 163–202. doi: 10.1007/s00285-012-0505-1.  Google Scholar

[45]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007) 1075–1081. doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[46]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern., 28 (1999), 665-683.   Google Scholar

show all references

References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.  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, Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.  Google Scholar

[3]

J. M. 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

[4]

J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169-179.  doi: 10.1007/BF01442552.  Google Scholar

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society, 63 (1977) 370–373. doi: 10.2307/2041821.  Google Scholar

[6]

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

[7]

H. T. Banks, P. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two-dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.  doi: 10.1093/ee/17.5.815.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

P. Cannarsa, G. Floridia and A. Y. 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

[14]

P. J. Davis, Interpolation and Approximation, Dover Publications, INC. New York, 1975.  Google Scholar

[15]

Z. Ditzian, Derivatives of Bernstein polynomials and smoothness, Proceedings of the American Mathematical Society, 93 (1985), 25-31.  doi: 10.1090/S0002-9939-1985-0766520-7.  Google Scholar

[16]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003.  Google Scholar

[17]

A. Doubova, E. Fernández-cara, Go nzález-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM. J. Control Optim, 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.  Google Scholar

[18]

L. C. Evans, Partial Differential Equation, Graduate Studies in Math, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[19]

C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[20]

E. Fern$\acute{a}$ndez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[21]

L. A. Fern$\acute{a}$ndez and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.  Google Scholar

[22]

L. A. Fern$\acute{a}$ndez 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

[23]

S. C. Ferreira, M. L. Martins and M. J. Vilela, Reaction-diffusion model for the growth of avascular tumor, Phys. Rev. E, 65 (2002), 021907. doi: 10.1103/PhysRevE.65.021907.  Google Scholar

[24]

A. Friedman, PDE problems arising in mathematical biology, Networks and Heterogeneousmedia, 7 (2012), 691-703.  doi: 10.3934/nhm.2012.7.691.  Google Scholar

[25]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes 34, Seoul National University, Korea, 1996.  Google Scholar

[26]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numerica, 3, (1994) 269–378. doi: 10.1017/S0962492900002452.  Google Scholar

[27]

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

[28]

A. 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

[29]

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

[30]

A. Y. Khapalov, Controllability of partial differential equations governed by multiplicative controls, Lecture Notes in Mathematics, 1995, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.  Google Scholar

[31]

K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems & Control Letters, 24 (1995), 301-306.  doi: 10.1016/0167-6911(94)00022-N.  Google Scholar

[32]

G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, Comm. PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[33]

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

[34]

G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Co., New York, 1986.  Google Scholar

[35]

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

[36]

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

[37]

M. Ouzahra, A. Tsouli and A. Boutoulout, Exact controllability of the heat equation with bilinear control, Mathematical Methods in the Applied Sciences, 38 (2015), 5074-5084.  doi: 10.1002/mma.3428.  Google Scholar

[38]

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

[39]

M. Ouzahra, Controllability of the semilinear wave equation governed by a multiplicative control, Evolution Equations & Control Theory, 8 (2019), 669-686.  doi: 10.3934/eect.2019039.  Google Scholar

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[41]

B. Perthame, Some mathematical aspects of tumor growth and therapy, Proceedings of the International Congress of Mathematicians—Seoul, Vol. 1, Kyung Moon Sa, Seoul, 2014,529–545.  Google Scholar

[42]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007) 179–208. doi: 10.1137/S0036144504446291.  Google Scholar

[43]

S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag Italia, Milan, 2008.  Google Scholar

[44]

A. Tosin, Initial/boundary-value problems of tumor growth within a host tissue, J. Math.Biol., 66 (2013) 163–202. doi: 10.1007/s00285-012-0505-1.  Google Scholar

[45]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007) 1075–1081. doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[46]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern., 28 (1999), 665-683.   Google Scholar

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