Optimal Merging Control of an Autonomous Vehicle in Mixed Traffic: an Optimal Index Policy

We consider the problem of a single Autonomous Vehicle (AV) merging into traffic consisting only of Human Driven Vehicles (HDVs) with the goal of minimizing both the travel time and energy consumption of the entire group of vehicles involved in the merging process. This is done by controlling only the AV and determining both the optimal merging sequence and the optimal AV trajectory associated with it. We derive an optimal index policy which prescribes the merging position of the AV within the group of HDVs. We also specify conditions under which the optimal index corresponds to the AV merging before all HDVs or after all HDVs, in which case no interaction of the AV with the HDVs is required. Simulation results are included to validate the optimal index policy and demonstrate cases where optimal merging can be achieved without requiring any explicit assumptions regarding human driving behavior.