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Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem
Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit
1. | Department of Astronautical Science and Mechanics, Harbin Institute of Technology, Harbin, China, China |
2. | Department of Mathematics and Statistics, Curtin University, Perth 6845 |
3. | Department of Mathematics and Statistics, Curtin University, Perth, Australia |
References:
[1] |
M. Bardi, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhauser, (1997).
doi: 10.1007/978-0-8176-4755-1. |
[2] |
L. D. Berkovitz, Necessary conditions for optimal strategies in a class of differential games and control problems,, SIAM Journal on Control and Optimization, 5 (1967), 1.
doi: 10.1137/0305001. |
[3] |
L. D. Berkovitz, The existence of value and saddle point in games of fixed duration,, SIAM Journal on Control and Optimization, 23 (1985), 172.
doi: 10.1137/0323015. |
[4] |
M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, part 2: Necessary conditions for optimal open-loop strategies,, Journal of Optimization Theory and Applications, 78 (1993), 443.
doi: 10.1007/BF00939877. |
[5] |
W. H. Clohessy and R. S. Wiltshire, Terminal guidance system for satellite rendezvous,, Journal of the Aerospace Sciences, 11 (1960), 653. Google Scholar |
[6] |
S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach,, Third Edition, (1981).
|
[7] |
K. Deb, A fast and elitist multi-objective genetic algorithm: NSGA-II,, IEEE Transactions on Evolutionary Computation, 6 (2002), 182. Google Scholar |
[8] |
A. Friedman, Differential Games,, American Mathematical Society, (1974).
|
[9] |
P. E. Gill, W. Murray, M. Saunders and M. H. Wright, User's Guide for NPSOL (Version 5.0): A Fortran Package for Nonlinear Programming,, Systems and Optimization Lab, (1998). Google Scholar |
[10] |
A. L. Herman and B. A. Conway, Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules,, Journal of Guidance, 19 (1996), 592.
doi: 10.2514/3.21662. |
[11] |
K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization,, Ph.D. Thesis, (2002). Google Scholar |
[12] |
K. Horie and B. A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization,, Journal of Guidance, 29 (2006), 105.
doi: 10.2514/1.3960. |
[13] |
R. Isaacs, Differential Games,, John Wiley and Sons, (1965).
|
[14] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual,, Department of Mathematics, (2002). Google Scholar |
[15] |
C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,, Journal of Optimization Theory and Applications, 154 (2012), 30.
doi: 10.1007/s10957-012-0006-9. |
[16] |
B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications,, ANZIAM Journal, 51 (2009), 162.
doi: 10.1017/S1446181110000040. |
[17] |
B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles,, Applied Mathematics and Computation, 224 (2013), 866.
doi: 10.1016/j.amc.2013.08.092. |
[18] |
B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem,, Journal of Optimization Theory and Applications, 151 (2011), 260.
doi: 10.1007/s10957-011-9904-5. |
[19] |
Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.
doi: 10.3934/jimo.2014.10.275. |
[20] |
R. C. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571.
doi: 10.3934/naco.2012.2.571. |
[21] |
R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.
doi: 10.1016/j.automatica.2009.05.029. |
[22] |
H. J. Oberle and W. Grimm, BNDSCO: A Program for the Numerical Solution of Optimal Control Problems,, Inst. für Angewandte Math. der Univ. Hamburg, (2001). Google Scholar |
[23] |
M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game,, Journal of Guidance, 31 (2008), 1111. Google Scholar |
[24] |
M. Pontani and B. A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game,, Journal of Guidance, 32 (2009), 474. Google Scholar |
[25] |
K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems,, Annals of Operations Research, 5 (1986), 485.
doi: 10.1007/BF02739235. |
[26] |
T. Shima and J. Shinar, Time-varying linear pursuit-evasion game models with bounded controls,, Journal of Optimization Theory and Applications, 25 (2002), 607.
doi: 10.2514/2.4927. |
[27] |
J. Shinar and T. Shima, Guidance law evaluation in highly nonlinear scenarios - comparison to linear analysis,, in Proceedings of the AIAA Guidance, (1999), 651.
doi: 10.2514/6.1999-4065. |
[28] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Third Edition, (2002).
doi: 10.1007/978-0-387-21738-3. |
[29] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991).
|
[30] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
show all references
References:
[1] |
M. Bardi, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhauser, (1997).
doi: 10.1007/978-0-8176-4755-1. |
[2] |
L. D. Berkovitz, Necessary conditions for optimal strategies in a class of differential games and control problems,, SIAM Journal on Control and Optimization, 5 (1967), 1.
doi: 10.1137/0305001. |
[3] |
L. D. Berkovitz, The existence of value and saddle point in games of fixed duration,, SIAM Journal on Control and Optimization, 23 (1985), 172.
doi: 10.1137/0323015. |
[4] |
M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuit-evasion type with state constraints, part 2: Necessary conditions for optimal open-loop strategies,, Journal of Optimization Theory and Applications, 78 (1993), 443.
doi: 10.1007/BF00939877. |
[5] |
W. H. Clohessy and R. S. Wiltshire, Terminal guidance system for satellite rendezvous,, Journal of the Aerospace Sciences, 11 (1960), 653. Google Scholar |
[6] |
S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach,, Third Edition, (1981).
|
[7] |
K. Deb, A fast and elitist multi-objective genetic algorithm: NSGA-II,, IEEE Transactions on Evolutionary Computation, 6 (2002), 182. Google Scholar |
[8] |
A. Friedman, Differential Games,, American Mathematical Society, (1974).
|
[9] |
P. E. Gill, W. Murray, M. Saunders and M. H. Wright, User's Guide for NPSOL (Version 5.0): A Fortran Package for Nonlinear Programming,, Systems and Optimization Lab, (1998). Google Scholar |
[10] |
A. L. Herman and B. A. Conway, Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules,, Journal of Guidance, 19 (1996), 592.
doi: 10.2514/3.21662. |
[11] |
K. Horie, Collocation with Nonlinear Programming for Two-Sided Flight Path Optimization,, Ph.D. Thesis, (2002). Google Scholar |
[12] |
K. Horie and B. A. Conway, Optimal fighter pursuit-evasion maneuvers found via two-sided optimization,, Journal of Guidance, 29 (2006), 105.
doi: 10.2514/1.3960. |
[13] |
R. Isaacs, Differential Games,, John Wiley and Sons, (1965).
|
[14] |
L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3 Optimal Control Software: Theory and User Manual,, Department of Mathematics, (2002). Google Scholar |
[15] |
C. Jiang, Q. Lin, C. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints,, Journal of Optimization Theory and Applications, 154 (2012), 30.
doi: 10.1007/s10957-012-0006-9. |
[16] |
B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications,, ANZIAM Journal, 51 (2009), 162.
doi: 10.1017/S1446181110000040. |
[17] |
B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles,, Applied Mathematics and Computation, 224 (2013), 866.
doi: 10.1016/j.amc.2013.08.092. |
[18] |
B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem,, Journal of Optimization Theory and Applications, 151 (2011), 260.
doi: 10.1007/s10957-011-9904-5. |
[19] |
Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.
doi: 10.3934/jimo.2014.10.275. |
[20] |
R. C. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results,, Numerical Algebra, 2 (2012), 571.
doi: 10.3934/naco.2012.2.571. |
[21] |
R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.
doi: 10.1016/j.automatica.2009.05.029. |
[22] |
H. J. Oberle and W. Grimm, BNDSCO: A Program for the Numerical Solution of Optimal Control Problems,, Inst. für Angewandte Math. der Univ. Hamburg, (2001). Google Scholar |
[23] |
M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game,, Journal of Guidance, 31 (2008), 1111. Google Scholar |
[24] |
M. Pontani and B. A. Conway, Numerical solution of the three-dimensional orbital pursuit-evasion game,, Journal of Guidance, 32 (2009), 474. Google Scholar |
[25] |
K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems,, Annals of Operations Research, 5 (1986), 485.
doi: 10.1007/BF02739235. |
[26] |
T. Shima and J. Shinar, Time-varying linear pursuit-evasion game models with bounded controls,, Journal of Optimization Theory and Applications, 25 (2002), 607.
doi: 10.2514/2.4927. |
[27] |
J. Shinar and T. Shima, Guidance law evaluation in highly nonlinear scenarios - comparison to linear analysis,, in Proceedings of the AIAA Guidance, (1999), 651.
doi: 10.2514/6.1999-4065. |
[28] |
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Third Edition, (2002).
doi: 10.1007/978-0-387-21738-3. |
[29] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems,, Longman Scientific and Technical, (1991).
|
[30] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
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