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

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  • 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}. $

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

    Citation:

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

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

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

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