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

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

Aeronautical Engineering: A continuing bibliography (supplement 158)

This bibliography lists 499 reports, articles and other documents introduced into the NASA scientific and technical information system in January 1983

Aeronautical Engineering: A continuing bibliography, supplement 96

This bibliography lists 448 reports, articles, and other documents introduced into the NASA scientific and technical information system in April 1978

A cumulative index to the 1973 issues of Aeronautical engineering: A special bibliography

This publication is a cumulative index to the abstracts contained in NASA SP-7037 (28) through NASA SP-7037 (39) of Aeronautical Engineering: A Special Bibliography. NASA SP-7037 and its supplements have been compiled through the cooperative efforts of the American Institute of Aeronautics and Astronautics (AIAA) and the National Aeronautics and Space Administration (NASA). This cumulative index includes subject, personal author, corporate source, contract, and report number indexes

A cumulative index to Aeronautical Engineering: A continuing bibliography

This bibliography is a cumulated index to the abstracts contained in NASA SP-7037(132) through NASA SP-7037(143) of Aeronautical Engineering: A continuing bibliography. NASA SP-7037 and its supplements have been compiled through the cooperative efforts of the American Institute of Aeronautics and Astronautics (AIAA) and the National Aeronautics and Space Administration (NASA). This cumulative index includes subject, personal author, corporate source, contract, and report number indexes

Aeronautical Engineering: A special bibliography with indexes, supplement 37

This special bibliography lists 511 reports, articles, and other documents introduced into the NASA scientific and technical information system in October, 1973

Aeronautical engineering: A continuing bibliography (supplement 152)

The bibliography lists 338 reports, articles and other documents introduced into the NASA scientific and technical information system in August 1982

Aeronautical engineering: A continuing bibliography with indexes (supplement 223)

This bibliography lists 423 reports, articles, and other documents introduced into the NASA scientific and technical information system in January, 1988

Cumulative index to NASA Tech Briefs, 1986-1990, volumes 10-14

Tech Briefs are short announcements of new technology derived from the R&D activities of the National Aeronautics and Space Administration. These briefs emphasize information considered likely to be transferrable across industrial, regional, or disciplinary lines and are issued to encourage commercial application. This cumulative index of Tech Briefs contains abstracts and four indexes (subject, personal author, originating center, and Tech Brief number) and covers the period 1986 to 1990. The abstract section is organized by the following subject categories: electronic components and circuits, electronic systems, physical sciences, materials, computer programs, life sciences, mechanics, machinery, fabrication technology, and mathematics and information sciences