Ordered Upwind Methods for Hybrid Control

We introduce a family of highly efficient (non-iterative) numerical methods for a wide class of hybrid control systems. The application of Dijkstra's classical method to a discrete optimal trajectory problem on a network obtains the solution in O(M log M) operations. The key idea behind the method is a careful use of the direction of information propagation, stemming from the optimality principle. In a series of recent papers, we have introduced a number of Ordered Upwind Methods (OUMs) to efficiently solve the fully anisotropic continuous optimal control problems. These techniques rely on using a partial information on the characteristic directions of the Hamilton-Jacobi-Bellman PDE, stemming from the continuous variant of the optimality principle. The resulting non-iterative algorithms have the computational complexity of O(M log M), where M is the total number of grid points where the solution is computed, regardless of the dimension of the control/state variables. In this paper, we showho wOrde red Upwind Methods may be extended to efficiently solve the hybrid (discrete/continuous) control problems. We illustrate our methods by solving a series of hybrid optimal trajectory problems with and without time-dependence of anisotropic speed functions.