Synthesis of Optimal Bang–Bang Control for Cooperative Collision Avoidance for Aircraft (Ships) with Unequal Linear Speeds

Close proximity encounters most often occur for situations in which participants have unequal linear speeds. Cooperative collision avoidance strategies for such situations are investigated. We show that, unlike the encounters of participants with equal linear speeds, bang–bang collision avoidance strategies are not always optimal when the linear speeds are unequal, and we establish the conditions for which no optimal bang–bang controls exist near the terminal time. Nevertheless, under certain conditions, we demonstrate that bang–bang collision avoidance strategies remain optimal for encounters of participants with unequal linear speeds. Such conditions are established, and it appears that they cover a wide range of important practical situations. The synthesis of bang–bang control is constructed, and its optimality is established.