Singular controls in optimal collision avoidance for participants with unequal linear speeds

Maurer, Helmut ; Tarnopolskaya, Tanya ; Fulton, Neale

ANZIAM Journal, Tome 52 (2012), / Harvested from Australian Mathematical Society

Résumé

This article studies optimal collision avoidance strategies for participants with unequal linear speeds in a planar close proximity encounter. It is known that bang-bang collision avoidance strategies are optimal for encounters of participants with equal linear speeds. However, as shown recently, bang-bang collision avoidance strategies are not necessarily optimal when the linear speeds of the participants are not equal. We study the structure of optimal singular controls for collision avoidance of participants with unequal linear speeds, but equal turn capabilities. We prove that both controls cannot be singular simultaneously, and that the only possible singular control is a zero control. We use several optimization techniques compute optimal state, control and adjoint variables. Numerical simulations suggest that a zero control strategy only exists for a slower participant and that, at most, one switching from a bang-bang to a singular control occurs. Different types of structural changes of the controls with change in the initial conditions are identified via the numerical simulations. References U. Ledzewicz, H. Maurer, and H. Schattler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8, 2011, 307--328. doi:10.3934/mbe.2011.8.307 U. Ledzewicz, H. Maurer, H. Schattler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 22, 2009, 13--26. doi:10.1016/j.mbs.2009.08.004 H. Maurer, C. Buskens, J.-H. R. Kim, Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optimal Control Applications and Methods, 26, 2005, 129--156. doi:10.1002/oca.756 A. W. Merz, Optimal aircraft collision avoidance. Proceedings of the Joint Automatic Control Conf., Paper 15-3, pages 449--454, 1973. A. W. Merz, Optimal evasive manoeuvres in maritime collision avoidance. Navigation, 20(2), 1973, 144--152. T. Tarnopolskaya, and N. L. Fulton, Optimal cooperative collision avoidance strategy for coplanar encounter: Merz's solution revisited, J. Optim. Theory Appl., 140 (2), 2009, 355--375. doi:10.1007/s10957-008-9452-9 T. Tarnopolskaya, and N. L. Fulton, Parametric behavior of the optimal control solution for collision avoidance in a close proximity encounter, In R. S. Andersson et al., editors, 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation, pages 425--431, 2009. http://www.mssanz.org.au/modsim09/A7/tarnopolskaya_A7.pdf T. Tarnopolskaya, and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance for aircraft (ships) with unequal turn capabilities, J. Optim. Theory Appl., 144 (2), 2010, 367--390. doi:10.1007/s10957-009-9597-1 T. Tarnopolskaya, and N. L. Fulton, Dispersal Curves for Optimal Collision Avoidance in a Close Proximity Encounter: a Case of Participants with Unequal Turn Rates, Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2010, WCE 2010, pages 1789--1794. IAENG, 2010. http://www.iaeng.org/publication/WCE2010/WCE2010_pp1789-1794.pdf T. Tarnopolskaya, and N. L. Fulton, Non-unique Optimal Collision Avoidance Strategies for Coplanar Encounter of Participants with Unequal Turn Capabilities, IAENG International Journal of Applied Mathematics, 40 (4), 2010, 289--296. http://www.iaeng.org/IJAM/issues_v40/issue_4/IJAM_40_4_10.pdf T. Tarnopolskaya, and N. L. Fulton, Synthesis of Optimal Control for Cooperative Collision Avoidance in a Close Proximity Encounter: Special Cases, Proceedings of the 18th World Congress of the International Federation of Automatic Control (IFAC), Milan (Italy), August 28--September 2, 2011, 18(1), 9775--9781. T. Tarnopolskaya, N. L. Fulton, and H. Maurer, Synthesis of optimal bang-bang control for cooperative collision avoidance for aircraft (ships) with unequal linear speeds. J. Optim. Theory Appl., to appear, 2012.

Détail

Publié le : 2012-01-01
DOI : https://doi.org/10.21914/anziamj.v53i0.5098