# American Institute of Mathematical Sciences

April  2014, 10(2): 443-460. doi: 10.3934/jimo.2014.10.443

## Computation of bang-bang and singular controls in collision avoidance

 1 Institute of Computational and Applied Mathematics, University of Muenster, Einsteinstr. 62, D-48149 Muenster, Germany 2 CSIRO Computational Informatics, Locked Bag 17, North Ryde NSW 1670, Australia 3 CSIRO Computational Informatics, GPO Box 664, Canberra ACT 2601, Australia

Received  November 2012 Revised  August 2013 Published  October 2013

We study optimal cooperative collision avoidance strategies for two participants in a planar close proximity encounter. Previous research focused on special cases of this problem and showed that bang-bang strategies without switching are optimal in most situations, while singular controls only appear for the case of participants with unequal linear speeds under certain conditions. This paper extends the earlier analyses to a general case of a coplanar close proximity encounter, for which both parameters of the problem may take arbitrary admissible values. For such a case, we present a theoretical and numerical study of the structure of optimal controls. 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. We identify different types of structural changes of the control strategies and show how the control structure changes with the change in the model parameters and initial conditions.
Citation: Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
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