Stabilization of nonlinear partial differential equations: theory and methods – StarPDE

Taking action to improve the world around us is an intrinsic aspect of human endeavors in many sciences. One mathematical approach consists in viewing the world as a system governed by differential equations (ordinary, partial or stochastic) and understanding how to act on it to attain a desired system state. This is the essence of control theory. In this context, one of the most practical questions is the stabilization problem: how can we act on the system to attain and maintain a desired state or trajectory that is naturally unstable?

The main characteristic of stabilization is that the control – i.e., the means to influence the system – depends on the state of the system. This serves both as one of its primary strengths and challenges, as this results in a feedback loop, which can lead to difficulties even only to ensure the problem’s well-posedness. While many generic methods have been successfully developed to address this issue when dealing with systems governed by ordinary differential equations, the stabilization of systems involving partial differential equations (PDEs) remains notably intricate, especially for nonlinear systems.

The purpose of this proposal is to develop novel mathematical approaches to tackle these limitations, with a specific focus on the following three challenges:

1. The general stabilization problem for PDEs: finding explicit and quantitative generic feedback laws in challenging frameworks, by relying on a new method, called generalized backstepping.

2. The stabilization of density-velocity systems – one of the richest classes of hyperbolic systems – with only local measurements. This has numerous applications in fluid mechanics and road traffic flow.

3. The design of machine learning techniques, based in particular on reinforcement learning and generative models, to help mathematicians solve open problems in stabilization, either by providing strong insights or by predicting a candidate solution.