Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats

Controlling systems of ordinary differential equations (ODEs) is ubiquitous in science and engineering. For finding an optimal feedback controller, the value function and associated fundamental equations such as the Bellman equation and the Hamilton-Jacobi-Bellman (HJB) equation are essential. The numerical treatment of these equations poses formidable challenges due to their non-linearity and their (possibly) high-dimensionality. In this paper we consider a finite horizon control system with associated Bellman equation. After a time-discretization, we obtain a sequence of short time horizon problems which we call local optimal control problems. For solving the local optimal control problems we apply two different methods, one being the well-known policy iteration, where a fixed-point iteration is required for every time step. The other algorithm borrows ideas from Model Predictive Control (MPC), by solving the local optimal control problem via open-loop control methods on a short time horizon, allowing us to replace the fixed-point iteration by an adjoint method. For high-dimensional systems we apply low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. We prove a linear error propagation with respect to the time discretization and give numerical evidence by controlling a diffusion equation with unstable reaction term and an Allen-Kahn equation.