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March  2020, 10(1): 157-169. doi: 10.3934/mcrf.2019034

Controllability properties of degenerate pseudo-parabolic boundary control problems

Mu-Ming Zhang , Tian-Yuan Xu , and  Jing-Xue Yin

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: Tian-Yuan Xu

Received  September 2018 Revised  February 2019 Published  April 2019

Fund Project: The first author is supported by the China Postdoctoral Science Foundation grant 2018M630960, and the Excellent Young Scholar Program of South China Normal University grant 17KJ01. The second author is supported by the Joint Training PhD Program of China Scholarship Council grant 201806750016, and the Innovation Project of Graduate School of South China Normal University grant 2018LKXM005. The third author is supported by NSFC grant 11771156.

This paper concerns with the boundary control of a degenerate pseudo-parabolic equation. Compare to the results those for degenerate parabolic equations, we discover that the null controllability property for the degenerate pseudo-parabolic equation is false, but the approximate controllability in some proper state space holds.

Keywords: Controllability, observability, degenerate pseudo-parabolic equation.
Mathematics Subject Classification: Primary: 93B05, 93B07; Secondary: 35K10.
Citation: Mu-Ming Zhang, Tian-Yuan Xu, Jing-Xue Yin. Controllability properties of degenerate pseudo-parabolic boundary control problems. Mathematical Control & Related Fields, 2020, 10 (1) : 157-169. doi: 10.3934/mcrf.2019034
References:
[1]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[2]

H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2.  Google Scholar

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C. Cances, C. Choquet, Y. Fan and I. S. Pop, Existence of weak solutions to a degenerate pseudo-parabolic equation modeling two-phase flow in porous media, CASA Report, (2010), 10–75, Available from: https://pure.tue.nl/ws/files/3187782/695284.pdf. Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[6]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, preprint, arXiv: 1801.01380. Google Scholar

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P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Mathematical Control and Related Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.  Google Scholar

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R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York, 1953.  Google Scholar

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C. M. Cuesta and J. Hulshof, A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves, Nonlinear Anal., 52 (2003), 1199-1218.  doi: 10.1016/S0362-546X(02)00160-8.  Google Scholar

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A. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), 65-83.  doi: 10.1016/S0377-0427(00)00635-X.  Google Scholar

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E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.  Google Scholar

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

[13]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.  Google Scholar

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M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

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A. Hasan, O. M. Aamo and B. Foss, Boundary control for a class of pseudo-parabolic differential equations, Systems Control Lett., 62 (2013), 63-69.  doi: 10.1016/j.sysconle.2012.10.009.  Google Scholar

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S. Ji, J. Yin and Y. Cao, Instability of Positive Periodic Solutions for Semilinear Pseudo-Parabolic Equations with Logarithmic Nonlinearity, J. Differential Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar

[17]

K. B. Liaskos, A. A. Pantelous and I. G. Stratis, Linear stochastic degenerate Sobolev equations and applications, Internat. J. Control, 88 (2015), 2538-2553.  doi: 10.1080/00207179.2015.1048482.  Google Scholar

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X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Optim., 50 (2012), 2046-2064.  doi: 10.1137/110851808.  Google Scholar

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Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[20]

M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675.  doi: 10.1016/j.na.2006.03.046.  Google Scholar

[21]

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Int. J. Tomogr. Stat., 5 (2007), 79-84.   Google Scholar

[22]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[23]

Q. Tao, H. Gao and Z. Yao, Null controllability of a pseudo-parabolic equation with moving control, J. Math. Anal. Appl., 418 (2014), 998-1005.  doi: 10.1016/j.jmaa.2014.04.038.  Google Scholar

[24]

C. J. Van Duijn, Y. Fan, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl., 14 (2013), 1361-1383.  doi: 10.1016/j.nonrwa.2012.10.002.  Google Scholar

[25]

L. W. White, Control of a pseudo-parabolic initial value problem to a target function, SIAM J. Control Optim., 17 (1979), 587-595.  doi: 10.1137/0317041.  Google Scholar

[26]

L. W. White, Controllability properties of pseudoparabolic boundary control problems, SIAM J. Control Optim., 18 (1980), 534-539.  doi: 10.1137/0318039.  Google Scholar

[27]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 4 (2010), 3008–3034. doi: 10.1007/978-0-387-89488-1.  Google Scholar

[28]

X. Zhang and E. Zuazua, The linearized Benjamin-Bona-Mahony equation: A spectral approach to unique continuation. Semigroups of operators: theory and applications, Optimization Software, (2002), 368-379.   Google Scholar

[29]

X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann., 325 (2003), 543-582.  doi: 10.1007/s00208-002-0391-8.  Google Scholar

show all references

References:
[1]

J. L. Bona and V. A. Dougalis, An initial and boundary value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522.  doi: 10.1016/0022-247X(80)90098-0.  Google Scholar

[2]

H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2.  Google Scholar

[3]

C. Cances, C. Choquet, Y. Fan and I. S. Pop, Existence of weak solutions to a degenerate pseudo-parabolic equation modeling two-phase flow in porous media, CASA Report, (2010), 10–75, Available from: https://pure.tue.nl/ws/files/3187782/695284.pdf. Google Scholar

[4]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.  doi: 10.1137/04062062X.  Google Scholar

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[6]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, preprint, arXiv: 1801.01380. Google Scholar

[7]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Mathematical Control and Related Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.  Google Scholar

[8]

R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York, 1953.  Google Scholar

[9]

C. M. Cuesta and J. Hulshof, A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves, Nonlinear Anal., 52 (2003), 1199-1218.  doi: 10.1016/S0362-546X(02)00160-8.  Google Scholar

[10]

A. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), 65-83.  doi: 10.1016/S0377-0427(00)00635-X.  Google Scholar

[11]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.  Google Scholar

[12]

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

[13]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.  Google Scholar

[14]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

[15]

A. Hasan, O. M. Aamo and B. Foss, Boundary control for a class of pseudo-parabolic differential equations, Systems Control Lett., 62 (2013), 63-69.  doi: 10.1016/j.sysconle.2012.10.009.  Google Scholar

[16]

S. Ji, J. Yin and Y. Cao, Instability of Positive Periodic Solutions for Semilinear Pseudo-Parabolic Equations with Logarithmic Nonlinearity, J. Differential Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017.  Google Scholar

[17]

K. B. Liaskos, A. A. Pantelous and I. G. Stratis, Linear stochastic degenerate Sobolev equations and applications, Internat. J. Control, 88 (2015), 2538-2553.  doi: 10.1080/00207179.2015.1048482.  Google Scholar

[18]

X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations, SIAM J. Control Optim., 50 (2012), 2046-2064.  doi: 10.1137/110851808.  Google Scholar

[19]

Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift, J. Funct. Anal., 260 (2011), 832-851.  doi: 10.1016/j.jfa.2010.10.018.  Google Scholar

[20]

M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675.  doi: 10.1016/j.na.2006.03.046.  Google Scholar

[21]

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping, Int. J. Tomogr. Stat., 5 (2007), 79-84.   Google Scholar

[22]

R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001.  Google Scholar

[23]

Q. Tao, H. Gao and Z. Yao, Null controllability of a pseudo-parabolic equation with moving control, J. Math. Anal. Appl., 418 (2014), 998-1005.  doi: 10.1016/j.jmaa.2014.04.038.  Google Scholar

[24]

C. J. Van Duijn, Y. Fan, L. A. Peletier and I. S. Pop, Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media, Nonlinear Anal. Real World Appl., 14 (2013), 1361-1383.  doi: 10.1016/j.nonrwa.2012.10.002.  Google Scholar

[25]

L. W. White, Control of a pseudo-parabolic initial value problem to a target function, SIAM J. Control Optim., 17 (1979), 587-595.  doi: 10.1137/0317041.  Google Scholar

[26]

L. W. White, Controllability properties of pseudoparabolic boundary control problems, SIAM J. Control Optim., 18 (1980), 534-539.  doi: 10.1137/0318039.  Google Scholar

[27]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 4 (2010), 3008–3034. doi: 10.1007/978-0-387-89488-1.  Google Scholar

[28]

X. Zhang and E. Zuazua, The linearized Benjamin-Bona-Mahony equation: A spectral approach to unique continuation. Semigroups of operators: theory and applications, Optimization Software, (2002), 368-379.   Google Scholar

[29]

X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann., 325 (2003), 543-582.  doi: 10.1007/s00208-002-0391-8.  Google Scholar

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