On the control of the linear Kuramoto−Sivashinsky equation∗
Revised: 8 June 2015
Accepted: 10 September 2015
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.
Mathematics Subject Classification: 35K35 / 93B05
Key words: Kuramoto−Sivashinky equation / parabolic equation / boundary control / internal control / null controllability / moment theory / Carleman estimates
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