Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds

We study optimal cooperative collision avoidance strategies for two participants with unequal linear speeds, but equal turn capabilities, in a planar close proximity encounter. Previous research showed that bang-bang strategies are optimal for participants with equal linear speeds. However, for participants with unequal linear speeds, bang-bang strategies are not necessarily optimal and, hence, singular control arcs may occur. We present a theoretical and numerical study of the structure of optimal controls with bang-bang and singular arcs. We prove that both controls can not be singular simultaneously and that the only possible singular control is a zero control. We derive formulas for the singular surfaces and verify that sufficient conditions hold for the computed extremal solutions. Different types of structural changes of the control strategies are identified.