Publications by Shengquan Xiang

Global controllability and stabilization of the wave Maps equation from a circle to a sphere

Continuing the investigations started in the recent work [12] on semi-global controllability and stabilization of the (1+1)-dimensional wave maps equation with spatial domain

S1

and target

Sk

, where semi-global refers to the

2π

-energy bound, we prov ...

2023

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Shengquan Xiang

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 Lapla ...

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

Semi-global controllability of a geometric wave equation

Joachim 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 ha ...

2022

Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Shengquan Xiang

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 ...

2021

Stabilization of the linearized water tank system

Shengquan Xiang

In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform st ...

2021

Small-time global stabilization of the viscous Burgers equation with three scalar controls

Shengquan Xiang

We construct explicit time-varying feedback laws leading to the global (null) stabilization in small time of the viscous Burgers equation with three scalar controls. Our feedback laws use first the quadratic transport term to achieve the small-time global ...

2021

Quantitative rapid and finite time stabilization of the heat equation

Shengquan Xiang

The null controllability of the heat equation is known for decades [21, 25, 34]. The finite time stabilizability of the one dimensional heat equation was proved by Coron-Nguyên [15], while the same question for high dimensional spaces remained widely open. ...

2020

Small-time local stabilization of the two dimensional incompressible Navier-Stokes equations

Shengquan Xiang

We provide explicit time-varying feedback laws that locally stabilize the two dimensional internal controlled incompressible Navier-Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as ...

2020

Boundary Stabilization of Focusing NLKG near Unstable Equilibria: Radial Case

Joachim Krieger, Shengquan Xiang

We investigate the stability and stabilization of the cubic focusing Klein-Gordon equation around static solutions on the closed ball in

\mathbb{R}^3

. First we show that the system is linearly unstable near the static solution

u\equiv1

for any dissipat ...

2020

Cost for a controlled linear KdV equation

Joachim Krieger, Shengquan Xiang

The controllability of the linearized KdV equation with right Neumann control is studied in the pioneering work of Rosier [25]. However, the proof is by contradiction arguments and the value of the observability constant remains unknown, though rich mathem ...

2019

Shengquan Xiang

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Global controllability and stabilization of the wave Maps equation from a circle to a sphere

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Semi-global controllability of a geometric wave equation

Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Stabilization of the linearized water tank system

Small-time global stabilization of the viscous Burgers equation with three scalar controls

Quantitative rapid and finite time stabilization of the heat equation

Small-time local stabilization of the two dimensional incompressible Navier-Stokes equations

Boundary Stabilization of Focusing NLKG near Unstable Equilibria: Radial Case

Cost for a controlled linear KdV equation

Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Semi-global controllability of a geometric wave equation

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Global controllability and stabilization of the wave Maps equation from a circle to a sphere

Stabilization of the linearized water tank system

Cost for a controlled linear KdV equation

Small-time global stabilization of the viscous Burgers equation with three scalar controls

Quantitative rapid and finite time stabilization of the heat equation

Small-time local stabilization of the two dimensional incompressible Navier-Stokes equations

Boundary Stabilization of Focusing NLKG near Unstable Equilibria: Radial Case