Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

Hamilton-Jacobi PDEs are a central object in optimal control and differential games, enabling the computation of controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss two novel computational methods for the approximation of high-dimensional HJ PDEs. In the first part of the talk, I will present a numerical method based on tensor decompositions. Such a compressed representation of the value function has a complexity that scales linearly with respect to the dimension of the control system, allowing the solution of control problems with over 100 states. In the second part of the talk, I will discuss the construction of a class of causality-free, data-driven methods which circumvent the numerical solution of the HJ PDE. I will address the generation of a synthetic dataset based on the use of representation formulas (such as Lax-Hopf or Pontryagin’s Maximum Principle), which is then fed into a high-dimensional sparse polynomial/ANN model for training. The use of representation formulas providing gradient information is fundamental to increase the data efficiency of the method. I will present applications in nonlinear dynamics, control of PDEs, and agent-based models.