  We introduce high order Bellman equations, extending classical Bellman
equations to the tensor setting. We introduce weakly chained diagonally
dominant (w.c.d.d.) tensors and show that a sufficient condition for the
existence and uniqueness of a positive solution to a high order Bellman
equation is that the tensors appearing in the equation are w.c.d.d. M-tensors.
In this case, we give a policy iteration algorithm to compute this solution. We
also prove that a weakly diagonally dominant Z-tensor with nonnegative
diagonals is a strong M-tensor if and only if it is w.c.d.d. This last point is
analogous to a corresponding result in the matrix setting and tightens a result
from [L. Zhang, L. Qi, and G. Zhou. "M-tensors and some applications." SIAM
Journal on Matrix Analysis and Applications (2014)]. We apply our results to
obtain a provably convergent numerical scheme for an optimal control problem
using an "optimize then discretize" approach which outperforms (in both
computation time and accuracy) a classical "discretize then optimize" approach.
To the best of our knowledge, a link between M-tensors and optimal control has
not been previously established.
