On the control of the linear Kuramoto−Sivashinsky equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 165-194.

In this paper we study the null controllability property of the linear Kuramoto−Sivashinsky equation by means of either boundary or internal controls. In the Dirichlet boundary case, we use the moment theory to prove that the null controllability property holds with only one boundary control if and only if the anti-diffusion parameter of the equation does not belong to a critical set of parameters. Regarding the Neumann boundary case, we prove that the null controllability property does not hold with only one boundary control. However, it does always hold when either two boundary controls or an internal control are considered. The proof of the latter is based on the controllability-observability duality and a suitable Carleman estimate.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2015044
Classification : 35K35, 93B05
Mots clés : Kuramoto−Sivashinky equation, parabolic equation, boundary control, internal control, null controllability, moment theory, Carleman estimates
Cerpa, Eduardo 1 ; Guzmán, Patricio 1 ; Mercado, Alberto 1

1 Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile.
@article{COCV_2017__23_1_165_0,
     author = {Cerpa, Eduardo and Guzm\'an, Patricio and Mercado, Alberto},
     title = {On the control of the linear {Kuramoto\ensuremath{-}Sivashinsky} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {165--194},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {1},
     year = {2017},
     doi = {10.1051/cocv/2015044},
     mrnumber = {3601020},
     zbl = {1364.35117},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2015044/}
}
TY  - JOUR
AU  - Cerpa, Eduardo
AU  - Guzmán, Patricio
AU  - Mercado, Alberto
TI  - On the control of the linear Kuramoto−Sivashinsky equation
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 165
EP  - 194
VL  - 23
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2015044/
DO  - 10.1051/cocv/2015044
LA  - en
ID  - COCV_2017__23_1_165_0
ER  - 
%0 Journal Article
%A Cerpa, Eduardo
%A Guzmán, Patricio
%A Mercado, Alberto
%T On the control of the linear Kuramoto−Sivashinsky equation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 165-194
%V 23
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2015044/
%R 10.1051/cocv/2015044
%G en
%F COCV_2017__23_1_165_0
Cerpa, Eduardo; Guzmán, Patricio; Mercado, Alberto. On the control of the linear Kuramoto−Sivashinsky equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 165-194. doi : 10.1051/cocv/2015044. http://www.numdam.org/articles/10.1051/cocv/2015044/

L. Baudouin, E. Cerpa, E. Crépeau and A. Mercado, Lipschitz stability in an inverse problem for the Kuramoto−Sivashinsky equation. Appl. Anal. 92 (2013) 2084–2102. | DOI | MR | Zbl

E. Cerpa, Null controllability and stabilization of the linear Kuramoto−Sivashinsky equation. Commun. Pure Appl. Anal. 9 (2010) 91–102. | DOI | MR | Zbl

E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto−Sivashinsky equation. J. Differ. Equ. 250 (2011) 2024–2044. | DOI | MR | Zbl

E. Cerpa, I. Rivas and B.-Y. Zhang, Boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM J. Control Optim. 51 (2013) 2976–3010. | DOI | MR | Zbl

J.-M. Coron, Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl

E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain. Differ. Integral Equ. 16 (2003) 303–326. | MR | Zbl

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Springer Verlag, Berlin, Heidelberg (2000). | MR | Zbl

H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equation in one space dimension. Arch. Rat. Mech. Anal. 43 (1971) 272–292. | DOI | MR | Zbl

A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Vol. 34 of Lecture Notes Series. Seoul National University, Korea (1996). | MR | Zbl

P. Gao, Insensitizing controls for the Cahn-Hilliard type equation. Electron. J. Qual. Theory Differ. Equ. 35 (2014) 1–22. | DOI | MR | Zbl

O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition. Systems Control Lett. 59 (2010) 390–395. | DOI | MR | Zbl

P. Guzmán 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 | MR | Zbl

C. Hu and R. Temam, Robust control of the Kuramoto−Sivashinsky equation. Dyn. Contin. Discrete Impuls. Syst. Ser. B 8 (2001) 315–338. | MR | Zbl

Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach. Prog. Theor. Phys. 54 (1975) 687–699. | DOI

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

W.-J. Liu and M. Krstić, Stability enhancement by boundary control in the Kuramoto−Sivashinsky equation. Nonlin. Anal. Ser. A 43 (2001) 485–507. | DOI | MR | Zbl

D.M. Michelson, G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames II: Numerical experiments. Acta Astronaut. 4 (1977) 1207–1221. | DOI | MR | Zbl

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Vol. 44 of Appl. Math. Sci. Springer Verlag, New York (1983). | MR | Zbl

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. | Numdam | MR | Zbl

G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames I: Derivation of basic equations. Acta Astronaut. 4 (1977) 1177–1206. | DOI | MR | Zbl

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser Verlag, Basel, Boston, Berlin (2009). | MR | Zbl

Z. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation. Taiwanese J. Math. 16 (2012) 1991–2017. | DOI | MR | Zbl

Cité par Sources :