30 nov 2023 -- 15:30   [open in google calendar]

Centro de Giorgi, Sala Conferenze

Abstract.

The classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. In this talk I will review some basics on optimal control problems and HJB equation and the challenges arising from its numerical approximation. The main disadvantage for this approach depends on the so-called curse of dimensionality, since the HJB equation and the dynamical system live in the same, possibly high dimensional, space. I will discuss possible strategies to tackle this problem. One approach is based on the construction of a tree structure taking into account all the possible trajectories and solving the HJB directly on this grid. This will guarantee a perfect matching with the discrete dynamics, allowing to derive rigorous error estimates. The second strategy is based on a low-rank approximation of the value function and the formulation of a supervised learning problem where the approximation is trained up control sampling. Finally, we will investigate a further reduction of the dimension by the use of Proper Orthogonal Decomposition, extracting the essential features of the system from a snapshot sequence of the trajectories.