### About me

I am a Tenure Track assistant professor at School of Mathematical Sciences, Peking University. Prior to that I was a Research Scientist (postdoc) at EPFL, in the group of Prof. Joachim Krieger.

Former student at École Normale Supérieure de Paris and Peking University, I obtained my Ph.D. on controllability and stabilization of fluids at Sorbonne Université under the supervision of Prof. Jean-Michel Coron, while in the year 2017 I was invited researcher at ETH Zürich.

You can contact me by shengquan.xiang[AT]pku.edu.cn

### Research

**Frequency Lyapunov for quantitative stabilization**

In [8, 9] I introduced the Frequency Lyapunov method, a constructive method that combines spectral inequalities and Lyapunov functionals, to get quantitative rapid stabilization, null-controllability with optimal costs, and finite time stabilization.

**Quantitative controllability and stabilization of dispersive equations**
Removing the compactness arguments for dispersive equations by quantitative approaches allows to construct more robust and applicable controls. In [2, 3, 6] we have studied constructive controllabilities for KdV equations describing waves in a canal. We quantitatively stabilise nonlinear waves equations with damping in [7, 15].

**Fredholm backstepping transformation**
In a series of works [10, 11, 14] we have investigated the Fredholm backstepping for a large class of operators, the compactness/duality method introduced in [14] overcomes the threshold imposed by the classical approach.

**Stabilization of systems emphasizing nonlinear effects**
In the works [1] and [4] we have benefited from nonlinear structures to stabilize the KdV equations and the viscous Burgers equation for which the linearized systems are not stabilizable.

**Control of the semiclassical Schrödinger equations**
In [12] we have combined the WKB method, the semiclassical limit and the geometrical nonlinear control techniques to get an approximate controllability of the quantum density and quantum momentum.

**Smoothing traffic flows with autonomous vehicles**
Traffic jams are generated from the instability of traffic equilibrium states [13], and increase strongly the fuel consumption and the emissions. We construct feedback laws on autonomous vehicles to stabilize these stop-and-go waves.

### Publications and preprints

19. Global controllability and stabilization of the wave maps equation from a circle to a sphere (with J.-M. Coron)

arXiv preprint 2024, submitted

18. Traffic Control via Connected and Automated Vehicles: An Open-Road Field Experiment with 100 CAVs (with A. Hayat, A. Bayen, B. Piccoli et. al. )

18. Traffic Control via Connected and Automated Vehicles: An Open-Road Field Experiment with 100 CAVs (with A. Hayat, A. Bayen, B. Piccoli et. al. )

arXiv preprint 2024, submitted

17. Traffic smoothing using explicit local controllers (with A. Hayat, A. Bayen, B. Piccoli et. al. )

17. Traffic smoothing using explicit local controllers (with A. Hayat, A. Bayen, B. Piccoli et. al. )

arXiv preprint 2023, submitted

16. Global controllability and stabilization of the wave maps equation from a circle to a sphere (with J.-M. Coron and J. Krieger)

16. Global controllability and stabilization of the wave maps equation from a circle to a sphere (with J.-M. Coron and J. Krieger)

arXiv preprint 2023, submitted

15. Semi-global controllability of a geometric wave equation (with J. Krieger)

15. Semi-global controllability of a geometric wave equation (with J. Krieger)

arXiv preprint 2022, submitted

14. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves (with L. Gagnon, A. Hayat and C. Zhang)

14. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves (with L. Gagnon, A. Hayat and C. Zhang)

to appear in

13. Stability of multi-population traffic flows (with A. Hayat and B. Piccoli)

12. On the global approximate controllability in small time of semiclassical 1-D Schrödinger equations between two states with positive quantum densities (with J.-M. Coron and P. Zhang)

11. Fredholm transformation on Laplacian and rapid stabilization for the heat equations (with L. Gagnon, A. Hayat and C. Zhang)

10. Stabilization of the linearized water tank system (with J.-M. Coron, A. Hayat and C. Zhang)

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

8. Quantitative rapid and finite time stabilization of the heat equation

7. Boundary stabilization of focusing NLKG near unstable equilibria: radial case (with J. Krieger)

6. Cost for a controlled linear KdV equation (with J. Krieger)

5. Stabilisation rapide d'équations de Burgers et de Korteweg-de Vries

*Annales de l’Institut Fourier,*79p.13. Stability of multi-population traffic flows (with A. Hayat and B. Piccoli)

*Networks and Heterogeneous Media*18 (2023), Issue 2: 877-905

12. On the global approximate controllability in small time of semiclassical 1-D Schrödinger equations between two states with positive quantum densities (with J.-M. Coron and P. Zhang)

*Journal of Differential Equations*345 (2023), 1-44

11. Fredholm transformation on Laplacian and rapid stabilization for the heat equations (with L. Gagnon, A. Hayat and C. Zhang)

*Journal of Functional Analysis*283 (2022), no.12, 67p.

10. Stabilization of the linearized water tank system (with J.-M. Coron, A. Hayat and C. Zhang)

*Archive for Rational Mechanics and Analysis*244 (2022), 1019–1097

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

*Annales de l’Institut Henri Poincaré, Analyse Non Linéaire*40 (2023), no. 6, 1487–1511

8. Quantitative rapid and finite time stabilization of the heat equation

*ESAIM: Control, Optimisation and Calculus of Variations*

7. Boundary stabilization of focusing NLKG near unstable equilibria: radial case (with J. Krieger)

*Pure and Applied Analysis*5 (2023), no. 4, 833–894

6. Cost for a controlled linear KdV equation (with J. Krieger)

*ESAIM: Control, Optimisation and Calculus of Variations*27 (2021) S21, 41p

5. Stabilisation rapide d'équations de Burgers et de Korteweg-de Vries

PhD. thesis 2019

4. Small-time global stabilization of the viscous Burgers equation with three scalar controls (with J.-M. Coron)

3. Null controllability of a linearized Korteweg-de Vries equation by backstepping approach

2. Small-time local stabilization for a Korteweg-de Vries equation

1. Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws (with J.-M. Coron and I. Rivas)

4. Small-time global stabilization of the viscous Burgers equation with three scalar controls (with J.-M. Coron)

*Journal de Mathématiques Pures et Appliquées*151 (7), 212-256, 2021

3. Null controllability of a linearized Korteweg-de Vries equation by backstepping approach

*SIAM J. Control Optim.*57 (2019), 1493–1515

2. Small-time local stabilization for a Korteweg-de Vries equation

*Systems & Control Letters*111 (2018), 64–69

1. Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws (with J.-M. Coron and I. Rivas)

*Analysis & PDE*10 (2017), no. 5, 1089–1122

### Teaching

**Spring 2023:**I will open a course on

*PDEs' Control Theory*