Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

Hernández-Santamaría, Víctor; Peralta, Liliana

doi:10.1007/s00028-022-00758-x

Cited by 6 publications

(6 citation statements)

References 27 publications

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“…by our initial assumption on null-controllability. Now, we will see that we can construct a nonzero terminal data (p T , q T ) such that identity (32) cannot hold. To this end, since Ω \ ω = ∅, there exists a point x 0 ∈ Ω\ω and therefore one can find a ball…”

Section: 3mentioning

confidence: 99%

“…Nonetheless, some spectral techniques have been applied to other simplifications of the SKS model (see e.g. [12,31,32]) and they can be a good starting point to extend the results in [19,20] to coupled systems of fourth-and second-order parabolic equations.…”

Section: Controllability Inmentioning

confidence: 99%

“…Although similar in formulation, changing the location of the control introduces several difficulties. Some results in this direction have been obtained in the stochastic setting in [32].…”

Section: Introductionmentioning

confidence: 99%

“…Although some of these techniques have been used for controlling coupled systems of fourth-and second-order parabolic equations at the continuous level (see e.g. [37,32]), their application for controlling time-discrete coupled systems is yet to be studied.…”

Section: Introductionmentioning

confidence: 99%

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

Hernández-Santamaría

2023

EECT

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<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \phi(\triangle t) $\end{document}</tex-math></inline-formula>-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter <inline-formula><tex-math id="M2">\begin{document}$ \triangle t $\end{document}</tex-math></inline-formula> 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.</p>

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Section: 3mentioning

confidence: 99%

Section: Controllability Inmentioning

confidence: 99%

“…Although similar in formulation, changing the location of the control introduces several difficulties. Some results in this direction have been obtained in the stochastic setting in [32].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Hernández-Santamaría

2023

EECT

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“…to (1.1) can be guaranteed by Proposition B.1. This system and some variants have been studied from a null-controllability point of view in several papers, see, for instance, [10,11,14,15,24].…”

mentioning

confidence: 99%

An insensitizing control problem for a linear Stabilized Kuramoto-Sivashinsky system

Bhandari¹,

Hernández-Santamaría²

2022

Preprint

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In this work, we address the existence of insensitizing controls for a coupled system of fourth-and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for control functions such that some functional of the state is locally insensitive to the perturbations of initial data. Let O be a nonempty observation set for the solution component(s) w.r.t. the L 2 -norm(s) and ω be another nonempty set where the interior controls are acting. Then, we study the associated insensitizing control problem under the assumption O ∩ ω = ∅ and as usual, it can be shown that this is equivalent to a null-controllability problem for a cascade system where the number of equations are doubled. This new problem is studied by means of the classical duality arguments and Carleman estimates, but unlike other insensitizing problems for scalar systems, the election of the Carleman tools depends on the number of components of the system to be insensitized.

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Boundary Controllability of a Simplified Stabilized Kuramoto-Sivashinsky System

Hernández-Santamaría,

Mercado,

Visconti

2023

Acta Appl Math

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Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

Cited by 6 publications

References 27 publications

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

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

An insensitizing control problem for a linear Stabilized Kuramoto-Sivashinsky system

Boundary Controllability of a Simplified Stabilized Kuramoto-Sivashinsky System

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