Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

Víctor Hernández-Santamaría

doi:10.3934/eect.2022038

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2023, Volume 12, Issue 2: 459-501. Doi: 10.3934/eect.2022038

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Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

Víctor Hernández-Santamaría

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C.P. 04510 CDMX, Mexico

Received: April 2021

Revised: October 2021

Early access: August 2022

Published: April 2023

The author is supported by the program "Estancias posdoctorales por México" of CONACyT, Mexico and by Project IN109522 of DGAPA-UNAM, Mexico

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Abstract

In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $ \phi(\triangle t) $-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $ \triangle t $ goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.

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Mathematics Subject Classification: Primary: 93C55, 93B05; Secondary: 35K52.

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