Calculus of Variations and Geometric Measure Theory

Numerical approximation of high-dimensional optimal control problems

Luca Saluzzi

created by malchiodi on 28 Nov 2023

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

Centro de Giorgi, Sala Conferenze


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.