An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems

We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This allows us to compute a neighborhood of the set of optimal trajectories, in order to reduce the search space. The solutions of both PDE are successively approximated by max-plus linear combinations of appropriate basis functions, using a hierarchy of finer and finer grids. We show that the sequence of approximate value functions obtained in this way does converge to the viscosity solution of the HJB equation in a neighborhood of optimal trajectories. Then, under certain regularity assumptions, we show that the number of arithmetic operations needed to compute an approximate optimal solution of a d-dimensional problem, up to a precision ε, is bounded by O(C^d|log ε|), for some constant C > 1, whereas ordinary grid-based methods have a complexity in O(1/ε^ad) for some constant a > 0.