### 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

**Random dynamical systems**

In [20] we introduced a criterion for the exponential mixing of random dynamical systems, which can be applied in particular to dispersive equation, and obtained the exponential mixing of the random nonlinear wave equations.

**Global control problems of geometric equations**
In [15, 16, 19] we initiated the research on the global control of geometric equations. Unlike the case with Euclidean space target, the geometric control problems rely heavily on topological and geometric properties of the Riemannian manifold target and nonlinear effects.

**Quantitative rapid stabilization**
By constructing explicit control we enhance the dissipation of the system, to make it decay as fast as we want. My contributions include the Fredholm backstepping for general operators [10, 11, 14], the Frequency Lyapunov method for parabolic equations [8, 9], and the first global type result [4].

**Dispersive equations and quantitative control**
The interest is based on the quantitative and constructive control approaches for dispersive equations, for example to remove the commonly used compactness arguments. This research topic includes the nonlinear wave equations with damping [7], the semiclassical Schrödinger equations [12], and KdV equations [1, 2, 3, 6].

**Smoothing traffic flows with autonomous vehicles (application)**
We designed algorithm on autonomous vehicles to stabilize stop-and-go waves [17, 18], which has been used in MegaVanderTest. This test deployed 100 vehicles in November of 2022, and was the largest coordinated open-road test to smooth traffic flow.

### Publications and preprints

21. Local large deviations for randomly forced nonlinear wave equations with localized damping
(with Y. Chen, Z. Liu, and Z. Zhang)

arXiv preprint 2024, submitted

20. Exponential mixing for random nonlinear wave equations: weak dissipation and localized control (with Z. Liu, D. Wei, J.-C. Zhao, and Z. Zhang)

20. Exponential mixing for random nonlinear wave equations: weak dissipation and localized control (with Z. Liu, D. Wei, J.-C. Zhao, and Z. Zhang)

arXiv preprint 2024, submitted

19. Global controllability to harmonic maps of the heat flow from a circle to a sphere (with J.-M. Coron)

19. Global controllability to harmonic maps of the heat flow 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 （In honor of Demetrios Christodoulou for his 70th birthday） (with J. Krieger)

15. Semi-global controllability of a geometric wave equation （In honor of Demetrios Christodoulou for his 70th birthday） (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

**Autumn 2024:**I will be teaching

*Linear Algebra B*