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Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative
Controllability properties of degenerate pseudo-parabolic boundary control problems
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
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.
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. |
[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. |
[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. |
[5] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.
|
[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. |
[8] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York, 1953. |
[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. |
[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. |
[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. |
[12] |
A. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Korea, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
L. Rosier and P. Rouchon,
On the controllability of a wave equation with structural damping, Int. J. Tomogr. Stat., 5 (2007), 79-84.
|
[22] |
R. E. Showalter and T. W. Ting,
Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[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. |
[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. |
[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. |
[26] |
L. W. White,
Controllability properties of pseudoparabolic boundary control problems, SIAM J. Control Optim., 18 (1980), 534-539.
doi: 10.1137/0318039. |
[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. |
[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.
|
[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. |
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. |
[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. |
[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. |
[5] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.
|
[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. |
[8] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York, 1953. |
[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. |
[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. |
[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. |
[12] |
A. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Korea, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
L. Rosier and P. Rouchon,
On the controllability of a wave equation with structural damping, Int. J. Tomogr. Stat., 5 (2007), 79-84.
|
[22] |
R. E. Showalter and T. W. Ting,
Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.
doi: 10.1137/0501001. |
[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. |
[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. |
[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. |
[26] |
L. W. White,
Controllability properties of pseudoparabolic boundary control problems, SIAM J. Control Optim., 18 (1980), 534-539.
doi: 10.1137/0318039. |
[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. |
[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.
|
[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. |
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