Hasnaa Zidani est Professeure à l'INSA Rouen Normandie.
Titre : Optimal Control problems ans Hamilton–Jacobi Approach in CAT(0) Spaces
Résumé : Optimal control problems are classically studied in smooth settings using the Hamilton–Jacobi–Bellman (HJB) framework. In this talk, we explore how these ideas can be extended to non-smooth geometric contexts. We focus on CAT(0) spaces, a broad class of geodesic metric spaces with non-positive curvature that naturally arise in applications such as networks, stratified domains, or spaces with junctions. These spaces provide a convenient setting to model situations where the underlying geometry is not smooth. After a brief overview of the HJB approach in the classical setting, we introduce a notion of viscosity solutions adapted to this metric framework. This allows us to characterize the value function of optimal control problems despite the absence of differentiable structure. We also discuss how controlled dynamics can be formulated using purely metric tools, leading to well-posed trajectories. Finally, we present some perspectives on numerical methods in this setting, along with simple examples illustrating the influence of the geometry on optimal trajectories.