Research Topics

• Controlled geometric flow of maps
  We initiated the research topic on the controlled geometric flows of maps, based on the heat flow and wave maps, where control acts as an intrinsic mechanism within geometric evolution equations. In contrast to classical Euclidean control theory, geometric and topological features of the target manifold directly shape the evolution and its controllability, leading to phenomena absent in flat settings. This perspective reveals new mechanisms governing short-time global controllability and yields structural characterizations, such as equivalences between controllability and homotopy classes, highlighting a fundamentally geometric nature of controlled dynamics.
  – Semi-global controllability of a geometric wave equation (with Krieger), PAMQ, 2024
  – Global controllability to harmonic maps of the heat flow from a circle to a sphere (with Coron), JMPA, 2025
  – Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy (with Coron, Krieger), preprint, 2025
• Structural aspects of controllability and observability for wave and dispersive equations
  This research line studies controllability and observability for wave and dispersive equations from a structural perspective, emphasizing the interplay between propagation, geometry, and nonlinearity. A central contribution is the discovery of the Observable Symmetry Condition in a spacetime framework, which provides a completion of the classical geometric control condition (GCC) for wave equations. These structural ideas extend naturally to KdV equations and semiclassical nonlinear Schrödinger equations.
  – Symmetry for the wave equation on torus: sharp unique continuation and observability conditions for spacetime regions (with Niu, Wang), preprint, 2025
  – Small-time local controllability of a KdV system for all critical lengths (with Niu), preprint, 2025
• Randomly forced wave and dispersive equations
  We formulated a control-oriented framework for the study of mixing and long-time statistical behavior in randomly forced wave and dispersive equations, and obtained results for NLW, NLS, and KdV. Within this framework, our contributions identify three core driving mechanisms: exponential asymptotic compactness arising from dispersive nonlinear smoothing; global dissipation induced by localized damping; and, localized control and quantitative stabilization playing a central role in the mixing mechanism.
  – Exponential mixing for random nonlinear wave equations: weak dissipation and localized control (with Liu, Wei, Zhang, Zhao), preprint, 2024
  – Exponential mixing for the randomly forced NLS equation (with Chen, Zhang, Zhao), preprint, 2025
• Quantitative stabilization
  This line emphasizes quantitative stabilization as a methodological perspective, focusing on how structural insights from control, spectral, and operator-theoretic analyses can be translated into effective and implementable feedback mechanisms. Representative contributions include the frequency Lyapunov method and related quantitative approaches to stabilization in nonlinear systems. This perspective also informs the methodology underlying the three research themes above.
  – Quantitative rapid and finite time stabilization of the heat equation, ESAIM: COCV, 2024
  – Fredholm backstepping for critical operators... (with Gagnon, Hayat, Zhang), AIF, 2025
• Application: autonomous vehicles for traffic smoothing
  These applications illustrate how the theoretical developments can be deployed in concrete models. We designed algorithm on autonomous vehicles to stabilize stop-and-go waves, which has been used in MegaVanderTest (Nov 2022). This test deployed 100 vehicles, and was the largest coordinated open-road test to smooth traffic flow.