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Person# Shengquan Xiang

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Heat equation

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was

Scalar field

In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensi

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the

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We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first prove that, under some assumptions on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. Then, we prove that the Fredholm transformation constructed for the Laplacian also leads to the local rapid stability of the viscous Burgers equation. (c) 2022 Elsevier Inc. All rights reserved.

We revisit the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with the Fredholm transformation upon Laplace operators in a sharp functional setting, which is the major objective of this work, from the Riesz basis properties and the operator equality to the stabilizing spaces. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.

2021Joachim Krieger, Shengquan Xiang

We prove the semi-global controllability and stabilization of the $(1+1)-$dimensional wave maps equation with spatial domain 𝕊1 and target $𝕊k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2π$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k=1$.

2022