  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.
