-
Previous Article
A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems
- MCRF Home
- This Issue
-
Next Article
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. |
[1] |
Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 |
[2] |
Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021005 |
[3] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020109 |
[4] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282 |
[5] |
Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 |
[6] |
Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021004 |
[7] |
Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020055 |
[8] |
Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052 |
[9] |
Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299 |
[10] |
Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054 |
[11] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[12] |
Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021004 |
[13] |
Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 |
[14] |
Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020444 |
[15] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[16] |
Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030 |
[17] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[18] |
João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 |
[19] |
Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297 |
[20] |
Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]