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Theory and applications of optimal control problems with multiple time-delays
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 |
References:
[1] |
M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals,, Numer. Algebra Control Optim., 2 (2012), 511.
doi: 10.3934/naco.2012.2.511. |
[2] |
M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm,, Ph.D thesis, (2011). Google Scholar |
[3] |
C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen,, Ph.D thesis, (1998). Google Scholar |
[4] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control. SQP-based direct discretization methods for practical optimal control problems,, J. Comput. Appl. Math., 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[5] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in Online Optimization of Large Scale Systems (eds. M. Gr\, (2001), 3.
|
[6] |
R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar |
[7] |
M. Hestens, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966).
|
[8] |
A. J. Krener, The high order maximum principle and its application to singular extremals,, SIAM J. Control and Optimization, 15 (1977), 256.
doi: 10.1137/0315019. |
[9] |
H. Maurer, Numerical solution of singular control problems using multiple shooting methods,, J. Optimization Theory and Applications, 18 (1976), 235.
doi: 10.1007/BF00935706. |
[10] |
H. Maurer, C. Büskens, J.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control Appl. Methods, 26 (2005), 129.
doi: 10.1002/oca.756. |
[11] |
H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach,, SIAM J. Control and Optim., 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[12] |
H. Maurer, T. Tarnopolskaya and N. L. Fulton, Singular controls in optimal collision avoidance for participants with unequal linear speeds,, ANZIAM J., 53 (2012). Google Scholar |
[13] |
H. Maurer, T. Tarnopolskaya and N. L. Fulton, Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds,, in 51st IEEE Conference on Decision and Control (CDC), (2012), 7697.
doi: 10.1109/CDC.2012.6426792. |
[14] |
A. W. Merz, Optimal aircraft collision avoidance,, in Proceedings of the Joint Automatic Control Conference, (1973), 15. Google Scholar |
[15] |
A. W. Merz, Optimal evasive manoeuvres in maritime collision avoidance,, Navigation, 20 (1973), 144. Google Scholar |
[16] |
H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems,, J. of Guidance, 13 (1990), 153.
doi: 10.2514/3.20529. |
[17] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control. Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, Advances in Control and Design, (2012).
doi: 10.1137/1.9781611972368. |
[18] |
N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. I. Main results,, Control and Cybernetics, 34 (2005), 927.
|
[19] |
L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1965). Google Scholar |
[20] |
T. Tarnopolskaya and N. L. Fulton, Optimal cooperative collision avoidance strategy for coplanar encounter: Merz's solution revisited,, J. Optim. Theory Appl., 140 (2009), 355.
doi: 10.1007/s10957-008-9452-9. |
[21] |
T. Tarnopolskaya and N. L. Fulton, Parametric behavior of the optimal control solution for collision avoidance in a close proximity encounter,, in 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation (eds. R. S. Andersson et al.), (2009), 425. Google Scholar |
[22] |
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 (2010), 367.
doi: 10.1007/s10957-009-9597-1. |
[23] |
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,, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2010, (2010), 1789. Google Scholar |
[24] |
T. Tarnopolskaya and N. L. Fulton, Non-unique optimal collision avoidance strategies for coplanar encounter of participants with unequal turn capabilities,, IAENG Int. J. Appl. Math., 40 (2010), 289.
|
[25] |
T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance in a close proximity encounter: Special cases,, in Proceedings of the 18th World Congress of the International Federation of Automatic Control (IFAC), (2011), 9775. Google Scholar |
[26] |
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., 155 (2012), 115.
doi: 10.1007/s10957-012-0049-y. |
[27] |
G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen,, Ph.D thesis, (2005). Google Scholar |
[28] |
G. Vossen, Switching time optimization for bang-bang and singular controls,, J. Optim. Theory Appl., 144 (2010), 409.
doi: 10.1007/s10957-009-9594-4. |
[29] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
show all references
References:
[1] |
M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic conditions for bang-singular extremals,, Numer. Algebra Control Optim., 2 (2012), 511.
doi: 10.3934/naco.2012.2.511. |
[2] |
M. S. Aronna, Second Order Analysis of Optimal Control Problems with Singular Arcs. Optimality Conditions and Shooting Algorithm,, Ph.D thesis, (2011). Google Scholar |
[3] |
C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen,, Ph.D thesis, (1998). Google Scholar |
[4] |
C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control. SQP-based direct discretization methods for practical optimal control problems,, J. Comput. Appl. Math., 120 (2000), 85.
doi: 10.1016/S0377-0427(00)00305-8. |
[5] |
C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems,, in Online Optimization of Large Scale Systems (eds. M. Gr\, (2001), 3.
|
[6] |
R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, Duxbury Press, (1993). Google Scholar |
[7] |
M. Hestens, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966).
|
[8] |
A. J. Krener, The high order maximum principle and its application to singular extremals,, SIAM J. Control and Optimization, 15 (1977), 256.
doi: 10.1137/0315019. |
[9] |
H. Maurer, Numerical solution of singular control problems using multiple shooting methods,, J. Optimization Theory and Applications, 18 (1976), 235.
doi: 10.1007/BF00935706. |
[10] |
H. Maurer, C. Büskens, J.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls,, Optimal Control Appl. Methods, 26 (2005), 129.
doi: 10.1002/oca.756. |
[11] |
H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach,, SIAM J. Control and Optim., 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[12] |
H. Maurer, T. Tarnopolskaya and N. L. Fulton, Singular controls in optimal collision avoidance for participants with unequal linear speeds,, ANZIAM J., 53 (2012). Google Scholar |
[13] |
H. Maurer, T. Tarnopolskaya and N. L. Fulton, Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds,, in 51st IEEE Conference on Decision and Control (CDC), (2012), 7697.
doi: 10.1109/CDC.2012.6426792. |
[14] |
A. W. Merz, Optimal aircraft collision avoidance,, in Proceedings of the Joint Automatic Control Conference, (1973), 15. Google Scholar |
[15] |
A. W. Merz, Optimal evasive manoeuvres in maritime collision avoidance,, Navigation, 20 (1973), 144. Google Scholar |
[16] |
H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems,, J. of Guidance, 13 (1990), 153.
doi: 10.2514/3.20529. |
[17] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control. Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, Advances in Control and Design, (2012).
doi: 10.1137/1.9781611972368. |
[18] |
N. P. Osmolovskii and H. Maurer, Equivalence of second order optimality conditions for bang-bang control problems. I. Main results,, Control and Cybernetics, 34 (2005), 927.
|
[19] |
L. S. Pontryagin, W. G. Boltyanski, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1965). Google Scholar |
[20] |
T. Tarnopolskaya and N. L. Fulton, Optimal cooperative collision avoidance strategy for coplanar encounter: Merz's solution revisited,, J. Optim. Theory Appl., 140 (2009), 355.
doi: 10.1007/s10957-008-9452-9. |
[21] |
T. Tarnopolskaya and N. L. Fulton, Parametric behavior of the optimal control solution for collision avoidance in a close proximity encounter,, in 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation (eds. R. S. Andersson et al.), (2009), 425. Google Scholar |
[22] |
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 (2010), 367.
doi: 10.1007/s10957-009-9597-1. |
[23] |
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,, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2010, (2010), 1789. Google Scholar |
[24] |
T. Tarnopolskaya and N. L. Fulton, Non-unique optimal collision avoidance strategies for coplanar encounter of participants with unequal turn capabilities,, IAENG Int. J. Appl. Math., 40 (2010), 289.
|
[25] |
T. Tarnopolskaya and N. L. Fulton, Synthesis of optimal control for cooperative collision avoidance in a close proximity encounter: Special cases,, in Proceedings of the 18th World Congress of the International Federation of Automatic Control (IFAC), (2011), 9775. Google Scholar |
[26] |
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., 155 (2012), 115.
doi: 10.1007/s10957-012-0049-y. |
[27] |
G. Vossen, Numerische Lösungsmethoden, Hinreichende Optimalitätsbedingungen und Sensitivitätsanalyse für Optimale Bang-Bang und Singuläre Steuerungen,, Ph.D thesis, (2005). Google Scholar |
[28] |
G. Vossen, Switching time optimization for bang-bang and singular controls,, J. Optim. Theory Appl., 144 (2010), 409.
doi: 10.1007/s10957-009-9594-4. |
[29] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
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