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

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 discrete time dynamics avoiding the use of a fixed space grid which is the bottleneck for high-dimensional 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 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.