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March  2020, 9(1): 181-191. doi: 10.3934/eect.2020002

Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation

1. 

State Key Laboratory of Automotive Simulaion and Control, Jilin University, Changchun 130012, China

2. 

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

* Corresponding author: Peng Gao

Received  October 2018 Revised  March 2019 Published  March 2020 Early access  August 2019

Fund Project: This work is supported by Foundation of State Key Laboratory of Automotive Simulation and Control and NSFC Grant (11601073).

In this paper, we establish the global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation by five controls: one control is the right member of the equation and is constant with respect to the space variable, the four others are the boundary controls.

Citation: Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations and Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002
References:
[1]

N. Carreño and P. Guzmán, On the cost of null controllability of a fourth order parabolic equation, Journal of Differential Equations, 261 (2016), 6485-6520.  doi: 10.1016/j.jde.2016.08.042.

[2]

E. CerpaA. Mercado and A. Pazoto, Null controllability of the stabilized Kuramoto-Sivashinsky system with one distributed control, SIAM J. Control Optim., 53 (2015), 1543-1568.  doi: 10.1137/130947969.

[3]

E. Cerpa, Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation, Commun. Pure Appl. Anal., 9 (2010), 91-102.  doi: 10.3934/cpaa.2010.9.91.

[4]

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.

[5]

E. CerpaP. Guzmán and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 165-194.  doi: 10.1051/cocv/2015044.

[6]

M. Chapouly, Global controllability of nonviscous and viscous Burgers type equations, SIAM J. Control Optim., 48 (2009), 1567-1599.  doi: 10.1137/070685749.

[7]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation, Communications in Contemporary Mathematics, 11 (2009), 495-521.  doi: 10.1142/S0219199709003454.

[8]

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. 

[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]

P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444.  doi: 10.1016/j.jde.2015.08.053.

[11]

P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Anal., 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023.

[12]

P. Gao, Null controllability of the viscous Camassa-Holm equation with moving control, Proc. Indian Acad. Sci. Math. Sci., 126 (2016), 99-108.  doi: 10.1007/s12044-015-0262-3.

[13]

P. Gao, Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 4 (2015), 281-296.  doi: 10.3934/eect.2015.4.281.

[14]

A. González 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.

[15]

P. Guzmán, Local exact controllability to the trajectories of the Cahn-Hilliard equation, Applied Mathematics & Optimization, (2015), 1–28.

[16]

A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-45.  doi: 10.1063/1.865160.

[17]

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

[18]

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

[19]

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.

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409.

[21]

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.

show all references

References:
[1]

N. Carreño and P. Guzmán, On the cost of null controllability of a fourth order parabolic equation, Journal of Differential Equations, 261 (2016), 6485-6520.  doi: 10.1016/j.jde.2016.08.042.

[2]

E. CerpaA. Mercado and A. Pazoto, Null controllability of the stabilized Kuramoto-Sivashinsky system with one distributed control, SIAM J. Control Optim., 53 (2015), 1543-1568.  doi: 10.1137/130947969.

[3]

E. Cerpa, Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation, Commun. Pure Appl. Anal., 9 (2010), 91-102.  doi: 10.3934/cpaa.2010.9.91.

[4]

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.

[5]

E. CerpaP. Guzmán and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 165-194.  doi: 10.1051/cocv/2015044.

[6]

M. Chapouly, Global controllability of nonviscous and viscous Burgers type equations, SIAM J. Control Optim., 48 (2009), 1567-1599.  doi: 10.1137/070685749.

[7]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries equation, Communications in Contemporary Mathematics, 11 (2009), 495-521.  doi: 10.1142/S0219199709003454.

[8]

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. 

[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]

P. Gao, A new global Carleman estimate for Cahn-Hilliard type equation and its applications, J. Differential Equations, 260 (2016), 427-444.  doi: 10.1016/j.jde.2015.08.053.

[11]

P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Anal., 139 (2016), 169-195.  doi: 10.1016/j.na.2016.02.023.

[12]

P. Gao, Null controllability of the viscous Camassa-Holm equation with moving control, Proc. Indian Acad. Sci. Math. Sci., 126 (2016), 99-108.  doi: 10.1007/s12044-015-0262-3.

[13]

P. Gao, Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 4 (2015), 281-296.  doi: 10.3934/eect.2015.4.281.

[14]

A. González 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.

[15]

P. Guzmán, Local exact controllability to the trajectories of the Cahn-Hilliard equation, Applied Mathematics & Optimization, (2015), 1–28.

[16]

A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids, 28 (1985), 37-45.  doi: 10.1063/1.865160.

[17]

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

[18]

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

[19]

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.

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409.

[21]

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.

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