Ensuring safety of automated vehicle (AV) control systems in multi-lane mixed-autonomy traffic is challenging. Control barrier functions (CBFs) represent a promising approach in which control inputs are filtered to guarantee forward-invariance of satisfaction of desired safety properties. This allows for balancing safety with performance, such as in the context of data-driven adaptive cruise control systems, which may otherwise be difficult to assure safety for. In real-world deployments to mixed-autonomy multi-lane traffic, however, external disturbances such as cut-in events can generate violations to the satisfaction of safety properties which would otherwise be met, such as maintaining a minimum time gap between vehicles. This work extends the design of CBFs for AVs by explicitly considering the effect of cut-in events. We show that a commonly proposed CBF designed to maintain time-gap cannot guarantee collision avoidance in the event of a cut-in. We show that when paired with a secondary CBF designed to maintain a positive space-gap through the use of higher-order CBFs via simple switching logic that both collisions can be avoided when cut-ins occur, and that over time the desired time-gap will be restored. Additionally, we present criteria for pole placement and string-stability of the AV when choosing CBF parameters. A series of numerical experiments are presented to demonstrate the main results.
