Course information

This course is structured into five main parts:

— Modeling and understanding front or interface propagation phenomena.
This first part focuses on the modeling of front or interface propagation phenomena. We begin with a simplified model assuming sufficiently regular interfaces, and then introduce a rigorous mathematical formulation for more complex interfaces. We explore the theoretical concepts underlying these phenomena and study the mechanisms driving the propagation.

— Analysis of first-order Hamilton–Jacobi equations.
In this section, we examine the nonlinear partial differential equations that arise in the modeling of propagation phenomena. We study in detail the appropriate concepts of solutions for such equations, with particular emphasis on existence, uniqueness, and regularity. We also investigate the connection between solutions of the mathematical model and the underlying propagation mechanisms.

— Numerical methods.
The third part of the course is devoted to numerical approaches for solving Hamilton–Jacobi equations and related optimal control problems. We introduce several numerical schemes, such as semi-Lagrangian methods, ENO and WENO schemes, and the Fast Marching Method (FMM). Stability, consistency, and convergence properties of these methods are also discussed.

— Learning-based approximation of HJB equations.
Before moving to applications, we introduce learning-based approximation techniques specifically designed for HJB equations. We emphasize the importance of incorporating the structure of the PDE—such as monotonicity, causality, and the viscosity-solution framework—into the learning architecture. We also discuss the limitations of generic black-box neural approaches and the need to preserve the qualitative behavior of HJB solutions when learning value functions.

— Application to trajectory planning.
Finally, we apply the concepts and methods developed throughout the course to trajectory planning. We show how front propagation techniques and HJB-based methods can be used to model and efficiently compute trajectories in complex environments with static or moving obstacles, with applications to robotics and autonomous navigation.