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Energy stability for thermo-viscous fluids with a fading memory heat flux
Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation
1. | School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China |
References:
[1] |
J. P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Masson, Paris, 1984. |
[2] |
O. Bodart, M. Gonzalez-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Diff. Eq., 29 (2004), 1017-1050.
doi: 10.1081/PDE-200033749. |
[3] |
E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation, J. Differential Equations, 250 (2011), 2024-2044.
doi: 10.1016/j.jde.2010.12.015. |
[4] |
L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-II. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486.
doi: 10.1016/0009-2509(86)80033-1. |
[5] |
M. Chen, Null controllability with constraints on the state for the linear Korteweg-de Vries equation, Archiv der Mathematik., 104 (2015), 189-199.
doi: 10.1007/s00013-015-0730-0. |
[6] |
P. Collet, J. P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys., 152 (1993), 203-214.
doi: 10.1007/BF02097064. |
[7] |
C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., 67 (1988), 197-226.
doi: 10.2307/2152750. |
[8] |
P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ, 35 (2014), 1-22.
doi: 10.2307/2152750. |
[9] |
P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147.
doi: 10.1016/j.na.2015.01.015. |
[10] |
A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E., 53 (1996), 3573-3578.
doi: 10.1103/PhysRevE.53.3573. |
[11] |
J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306.
doi: 10.1002/cpa.3160470304. |
[12] |
P. G. Meléndez, Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto-Sivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290.
doi: 10.1016/j.jmaa.2013.05.050. |
[13] |
A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-245.
doi: 10.1063/1.865160. |
[14] |
M. S. Jolly, I. G. Kevrekidis and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Phys. D, 44 (1990), 38-60.
doi: 10.1016/0167-2789(90)90046-R. |
[15] |
Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. |
[16] |
Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys, 64 (1978), 346-367.
doi: 10.1143/PTPS.64.346. |
[17] |
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
doi: 10.1143/PTP.55.356. |
[18] |
C. Louis-Rose, A null controllability problem with a finite number of constraints on the normal derivative for the semilinear heat equation, Electron. J. Qual. Theory Differ. Equ., 95 (2012), 1-34. |
[19] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. |
[20] |
J. L. Lions, Sentinelles Pour Les Systèmes Distribués à Données Incomplètes, Masson, Paris, 1992. |
[21] |
R. E. Laquey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394.
doi: 10.1103/PhysRevLett.34.391. |
[22] |
G. M. Mophou, Null controllability with constraints on the state for nonlinear heat equations, Forum Math., 23 (2011), 285-319.
doi: 10.1515/FORM.2011.010. |
[23] |
G. M. Mophou and O. Nakoulima, Null controllability with constraints on the state for the semilinear heat equation, J. Optim. Theory Appl., 143 (2009), 539-565.
doi: 10.1007/s10957-009-9568-6. |
[24] |
O. Nakoulima, Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels, ESAIM Control Optim. Calc. Var., 13 (2007), 623-638.
doi: 10.1051/cocv:2007038. |
[25] |
B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Diff. Eq., 14 (1989), 245-297.
doi: 10.1080/03605308908820597. |
[26] |
S. Somdouda and G. M. Mophou, Null controllability with constraints on the state for the age-dependent linear population dynamics problem, Adv. Differ. Equ. Control Process., 10 (2012), 113-130. |
[27] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[28] |
G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[29] |
R. Temam and X. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case, Differential Integral Equations, 7 (1994), 1095-1108. |
[30] |
Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017. |
show all references
References:
[1] |
J. P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Masson, Paris, 1984. |
[2] |
O. Bodart, M. Gonzalez-Burgos and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Diff. Eq., 29 (2004), 1017-1050.
doi: 10.1081/PDE-200033749. |
[3] |
E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation, J. Differential Equations, 250 (2011), 2024-2044.
doi: 10.1016/j.jde.2010.12.015. |
[4] |
L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surfaces-II. Bifurcation analyses of the long-wave equation, Chem. Eng. Sci., 41 (1986), 2477-2486.
doi: 10.1016/0009-2509(86)80033-1. |
[5] |
M. Chen, Null controllability with constraints on the state for the linear Korteweg-de Vries equation, Archiv der Mathematik., 104 (2015), 189-199.
doi: 10.1007/s00013-015-0730-0. |
[6] |
P. Collet, J. P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys., 152 (1993), 203-214.
doi: 10.1007/BF02097064. |
[7] |
C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., 67 (1988), 197-226.
doi: 10.2307/2152750. |
[8] |
P. Gao, Insensitizing controls for the Cahn-Hilliard type equation, Electron. J. Qual. Theory Differ. Equ, 35 (2014), 1-22.
doi: 10.2307/2152750. |
[9] |
P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147.
doi: 10.1016/j.na.2015.01.015. |
[10] |
A. Gonzalez and A. Castellanos, Nonlinear electrohydrodynamic waves on films falling down an inclined plane, Phys. Rev. E., 53 (1996), 3573-3578.
doi: 10.1103/PhysRevE.53.3573. |
[11] |
J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306.
doi: 10.1002/cpa.3160470304. |
[12] |
P. G. Meléndez, Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto-Sivashinsky type equation, J. Math. Anal. Appl., 408 (2013), 275-290.
doi: 10.1016/j.jmaa.2013.05.050. |
[13] |
A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-245.
doi: 10.1063/1.865160. |
[14] |
M. S. Jolly, I. G. Kevrekidis and E. S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Phys. D, 44 (1990), 38-60.
doi: 10.1016/0167-2789(90)90046-R. |
[15] |
Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. |
[16] |
Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys, 64 (1978), 346-367.
doi: 10.1143/PTPS.64.346. |
[17] |
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.
doi: 10.1143/PTP.55.356. |
[18] |
C. Louis-Rose, A null controllability problem with a finite number of constraints on the normal derivative for the semilinear heat equation, Electron. J. Qual. Theory Differ. Equ., 95 (2012), 1-34. |
[19] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. |
[20] |
J. L. Lions, Sentinelles Pour Les Systèmes Distribués à Données Incomplètes, Masson, Paris, 1992. |
[21] |
R. E. Laquey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394.
doi: 10.1103/PhysRevLett.34.391. |
[22] |
G. M. Mophou, Null controllability with constraints on the state for nonlinear heat equations, Forum Math., 23 (2011), 285-319.
doi: 10.1515/FORM.2011.010. |
[23] |
G. M. Mophou and O. Nakoulima, Null controllability with constraints on the state for the semilinear heat equation, J. Optim. Theory Appl., 143 (2009), 539-565.
doi: 10.1007/s10957-009-9568-6. |
[24] |
O. Nakoulima, Optimal control for distributed systems subject to null-controllability. Application to discriminating sentinels, ESAIM Control Optim. Calc. Var., 13 (2007), 623-638.
doi: 10.1051/cocv:2007038. |
[25] |
B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Diff. Eq., 14 (1989), 245-297.
doi: 10.1080/03605308908820597. |
[26] |
S. Somdouda and G. M. Mophou, Null controllability with constraints on the state for the age-dependent linear population dynamics problem, Adv. Differ. Equ. Control Process., 10 (2012), 113-130. |
[27] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[28] |
G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames-I Derivation of basic equations, Acta Astronaut., 4 (1977), 1177-1206.
doi: 10.1016/0094-5765(77)90096-0. |
[29] |
R. Temam and X. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto-Sivashinsky equation in the general case, Differential Integral Equations, 7 (1994), 1095-1108. |
[30] |
Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017. |
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