Research

PDE control across geometry,
propagation, and randomness

To what extent can we shape the evolution of systems described by partial differential equations, and how much can we learn about them from limited observations? My research seeks to understand the structure of complex dynamical systems through control and observation, and to explore how control ideas reveal the mathematical principles underlying geometry, propagation, and randomness.

Controlled geometric flows 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.

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 introduction of the Observable Symmetry Condition in a spacetime framework, which identifies an additional structure complementing the classical geometric control condition (GCC) for wave equations.

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.

Quantitative stabilization
and its applications

How can we design feedback controls that stabilize a system at a prescribed rate, and what control cost does this require? I develop quantitative and constructive approaches to stabilization, with particular attention to the relationship between convergence rates and control costs. These methods also provide tools for the questions of controllability and long-time dynamics discussed above. Applications include traffic smoothing with autonomous vehicles.

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, Fredholm backstepping, and related quantitative approaches to stabilization in nonlinear systems.

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.

  • Energy-Saving Control of Freeway Traffic: Field Experiments and Results (with McQuade, Hayat, Piccoli, Bayen, et al.), revision resubmitted to Nature
  • Traffic smoothing via connected & automated vehicles: A modular, hierarchical control design deployed in a 100-cav flow smoothing experiment (with Bayen, Hayat, Piccoli et al.), IEEE Control Systems Magazine, 2025