Error Estimates for a Tree Structure Algorithm Solving Finite Horizon Control Problems^*

¹ Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK
² Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, Via Torino 155, 30172 Venezia Mestre, Italy
³ Sapienza Università di Roma – Piazzale Aldo Moro 5, 00185 Roma, Italy

^** Corresponding author: l.saluzzi@imperial.ac.uk

Received: 4 June 2021
Accepted: 13 October 2022

Abstract

In the dynamic programming approach to optimal control problems a crucial role is played by the value function that is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well known that this approach suffers from the “curse of dimensionality” and this limitation has reduced its use in real world applications. Here, we analyze a dynamic programming algorithm based on a tree structure to mitigate the “curse of dimensionality”. The tree is built by the discrete time dynamics avoiding the use of a fixed space grid which is the bottleneck for highdimensional problems, this also drops the projection on the grid in the approximation of the value function. In this work, we present first order error estimates for the the approximation of the value function based on the tree-structure algorithm. The estimate turns out to have the same order of convergence of the numerical method used for the approximation of the dynamics. Furthermore, we analyze a pruning technique for the tree to reduce the complexity and minimize the computational effort. Finally, we present some numerical tests to show the theoretical results.