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.
Mots-clés : Kuramoto−Sivashinky equation, parabolic equation, boundary control, internal control, null controllability, moment theory, Carleman estimates
@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/cocv/2015044/
, , and , Lipschitz stability in an inverse problem for the Kuramoto−Sivashinsky equation. Appl. Anal. 92 (2013) 2084–2102. | DOI | MR | Zbl
, Null controllability and stabilization of the linear Kuramoto−Sivashinsky equation. Commun. Pure Appl. Anal. 9 (2010) 91–102. | DOI | MR | Zbl
and , Local exact controllability to the trajectories of the 1-D Kuramoto−Sivashinsky equation. J. Differ. Equ. 250 (2011) 2024–2044. | DOI | MR | Zbl
, and , 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
, 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
and , 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
, Insensitizing controls for the Cahn-Hilliard type equation. Electron. J. Qual. Theory Differ. Equ. 35 (2014) 1–22. | DOI | MR | Zbl
and , Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition. Systems Control Lett. 59 (2010) 390–395. | DOI | MR | Zbl
, 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
and , Robust control of the Kuramoto−Sivashinsky equation. Dyn. Contin. Discrete Impuls. Syst. Ser. B 8 (2001) 315–338. | MR | Zbl
and , On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach. Prog. Theor. Phys. 54 (1975) 687–699. | DOI
and , Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55 (1976) 356–369. | DOI
and , Stability enhancement by boundary control in the Kuramoto−Sivashinsky equation. Nonlin. Anal. Ser. A 43 (2001) 485–507. | DOI | MR | Zbl
, , 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
, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. | Numdam | MR | Zbl
, 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
, 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 :
