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September  2015, 4(3): 281-296. doi: 10.3934/eect.2015.4.281

Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation

Peng Gao 1,

1. 

School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Received  March 2015 Revised  June 2015 Published  September 2015

This paper is addressed to study the null controllability with constraints on the state for the Kuramoto-Sivashinsky equation. We first consider the linearized problem. Then, by Kakutani fixed point theorem, we show that the same result holds for the Kuramoto-Sivashinsky equation.
Keywords: 1-D Kuramoto-Sivashinsky equation, Null controllability, constraints on the state..
Mathematics Subject Classification: Primary: 35K35, 35K55; Secondary: 93B0.
Citation: Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281
References:
[1]

J. P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Masson, Paris, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. Google Scholar

[16]

Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys, 64 (1978), 346-367. doi: 10.1143/PTPS.64.346.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[20]

J. L. Lions, Sentinelles Pour Les Systèmes Distribués à Données Incomplètes, Masson, Paris, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017.  Google Scholar

show all references

References:
[1]

J. P. Aubin, L'analyse Non Linéaire et ses Motivations Économiques, Masson, Paris, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Goodman, Stability of the Kuramoto-Sivashinsky and related systems, Comm. Pure Appl. Math., 47 (1994), 293-306. doi: 10.1002/cpa.3160470304.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Theor. Phys., 54 (1975), 687-699. Google Scholar

[16]

Y. Kuramoto, Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys, 64 (1978), 346-367. doi: 10.1143/PTPS.64.346.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[20]

J. L. Lions, Sentinelles Pour Les Systèmes Distribués à Données Incomplètes, Masson, Paris, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

Z. C. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math., 16 (2012), 1991-2017.  Google Scholar

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