Optimistic planning for control of hybrid-input nonlinear systems.

We propose two branch-and-bound, optimistic planning algorithms for discrete-time nonlinear optimal control problems in which there is a continuous and a discrete action (input). The dynamics and re-wards (negative costs) must be Lipschitz but can otherwise be general, as long as certain boundedness conditions are satisfied by the continuous action, reward, and Lipschitz constant of the dynamics. We start by investigating the structure of the space of hybrid-input sequences. Based on this structure, we propose for the first algorithm an optimistic selection rule that picks for refinement (branching) the subset with the largest upper bound on the value. At the price of a higher budget, the second method reduces the reliance on the Lipschitz constant, by refining all sets that are potentially optimistic. This effectively means that the Lipschitz constant is automatically optimized. The way to select the largest -impact action along which to refine the sets is the same for both algorithms, and still depends on the Lipschitz constant. We provide convergence rate guarantees for both methods, which link the computational budget to the near-optimality of the action sequences returned, in a way that depends on a problem complexity measure. We also give empirical results for a nonlinear problem, where the algorithms are applied in receding horizon, and depending on the budget either one or the other algorithm works better.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).