Towards hybrid optimal control

In this article a general class of hybrid optimal control problems with continuous and discrete state variables and control inputs is defined. After a brief review of conventional optimal control, major novel challenges resulting from the hybrid nature are discussed. Some application problems are comparatively easy to solve because of the fixed or known sequence of discrete events; however, if the number and the sequence of discrete phases is not known a priori, the solution must then be found among a combinatorial number of possible sequence candidates. The article presents several preliminary approaches to the (numerical) solution of hybrid optimal control problems by hybrid dynamic programming, by decomposition using branch-and-bound, or fixing transversality conditions to obtain suboptimal solutions. The last two methods rely on the capabilities of the direct collocation method DIRCOL to solving multiphase optimal control problems robustly and efficiently. Results obtained by the proposed methods are presented in two examples: an underactuated robotic system with a holding brake as the discrete component, and a hybrid, motorized traveling salesman problem.