Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

Francisco J. Vielma leal; Ademir Pastor

doi:10.3934/eect.2021062

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2022, Volume 11, Issue 5: 1745-1773. Doi: 10.3934/eect.2021062

This issue Previous Article Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations Next Article

Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

Francisco J. Vielma leal^, and
Ademir Pastor

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas-SP, Brazil

^* Corresponding author
^* Corresponding author

Received: July 31, 2021

Early access: December 2021

Published: October 2022

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Abstract

In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in $ H_{p}^{s}(\mathbb{T}) $ with $ s\in \mathbb{R}. $ We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in $ H_{p}^{s}(\mathbb{T}) $ with $ s\in \mathbb{R}. $

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Mathematics Subject Classification: Primary: 93B05, 93D15, 35J10, 37L50.

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Figure 1. Dispersion of $ \lambda_{k} $'s for KdV, Benjamin-Ono and Benjamin equations

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Figure 2. Dispersion of $ \lambda_{k} $'s for Schrödinger equation

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Figure 3. Dispersion of $ \lambda_{k} $'s for the Smith and fourth-order Schrödinger equations with $ \mu>0. $

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