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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 and 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.

[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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