Abstract:
Control theory explores the question: "If I can act on a system, what can I make it do ?" One of the central challenges within this field is the stabilization problem, which investigates how to manipulate certain aspects of an equation to ensure a desired long-term behavior in all solutions. This talk will highlight recent progress in the stabilization of partial differential equations (PDEs).
We will begin by introducing a novel approach known as Fredholm backstepping (or F-equivalence). Instead of directly attempting to solve the problem, this method involves selecting a control that allows the PDE system to be invertibly transformed into a simpler PDE system whose stability is already established. Significant advancements in the past two years have demonstrated the suprisingly remarkable efficacy of this technique.
Next, we will discuss a practical application: stabilizing hyperbolic PDEs to control traffic flow. We will show how abstract mathematical concepts, like entropic solutions, can have tangible impacts in real-world scenarios.
Lastly, if time allows, we will discuss recent advances in AI for mathematics: how to teach mathematical intuition to an AI to solve mathematical problems.
