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Second order sufficient conditions for a class of bilevel programs with lower level secondorder cone programming problem
Numerical solution of a pursuitevasion 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 HamiltonJacobiBellman Equations, Birkhauser, Boston, 1997. doi: 10.1007/9780817647551. 
[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), 124. 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), 172196. doi: 10.1137/0323015. 
[4] 
M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuitevasion type with state constraints, part 2: Necessary conditions for optimal openloop strategies, Journal of Optimization Theory and Applications, 78 (1993), 443463. 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), 653658. 
[6] 
S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third Edition, McGrawHill, New York, 1981. 
[7] 
K. Deb, A fast and elitist multiobjective genetic algorithm: NSGAII, IEEE Transactions on Evolutionary Computation, 6 (2002), 182197. 
[8] 
A. Friedman, Differential Games, American Mathematical Society, Rhode Island, 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, Stanford University, California, 1998. 
[10] 
A. L. Herman and B. A. Conway, Direct optimization using collocation based on highorder GaussLobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592599. doi: 10.2514/3.21662. 
[11] 
K. Horie, Collocation with Nonlinear Programming for TwoSided Flight Path Optimization, Ph.D. Thesis, University of Illinois at UrbanaChampaign, Champaign, 2002. 
[12] 
K. Horie and B. A. Conway, Optimal fighter pursuitevasion maneuvers found via twosided optimization, Journal of Guidance, Control, and Dynamics, 29 (2006), 105112. doi: 10.2514/1.3960. 
[13] 
R. Isaacs, Differential Games, John Wiley and Sons, New York, 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, The University of Western Australia, 2002. 
[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), 3053. doi: 10.1007/s1095701200069. 
[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), 162177. 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), 866875. 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), 260291. doi: 10.1007/s1095701199045. 
[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), 275309. 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, Control, and Optimization, 2 (2012), 571599. 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), 22502257. 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. 
[23] 
M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game, Journal of Guidance, Control, and Dynamics, 31 (2008), 11111122. 
[24] 
M. Pontani and B. A. Conway, Numerical solution of the threedimensional orbital pursuitevasion game, Journal of Guidance, Control, and Dynamics, 32 (2009), 474487. 
[25] 
K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems, Annals of Operations Research, 5 (1986), 485500. doi: 10.1007/BF02739235. 
[26] 
T. Shima and J. Shinar, Timevarying linear pursuitevasion game models with bounded controls, Journal of Optimization Theory and Applications, 25 (2002), 607618. 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, Navigation, and Control Conference, (1999), 651661. doi: 10.2514/6.19994065. 
[28] 
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third Edition, Springer, New York, 2002. doi: 10.1007/9780387217383. 
[29] 
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 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), 705718. doi: 10.3934/jimo.2009.5.705. 
show all references
References:
[1] 
M. Bardi, Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations, Birkhauser, Boston, 1997. doi: 10.1007/9780817647551. 
[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), 124. 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), 172196. doi: 10.1137/0323015. 
[4] 
M. Breitner, H. Pesch and W. Grimm, Complex differential games of pursuitevasion type with state constraints, part 2: Necessary conditions for optimal openloop strategies, Journal of Optimization Theory and Applications, 78 (1993), 443463. 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), 653658. 
[6] 
S. D. Conte and C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third Edition, McGrawHill, New York, 1981. 
[7] 
K. Deb, A fast and elitist multiobjective genetic algorithm: NSGAII, IEEE Transactions on Evolutionary Computation, 6 (2002), 182197. 
[8] 
A. Friedman, Differential Games, American Mathematical Society, Rhode Island, 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, Stanford University, California, 1998. 
[10] 
A. L. Herman and B. A. Conway, Direct optimization using collocation based on highorder GaussLobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592599. doi: 10.2514/3.21662. 
[11] 
K. Horie, Collocation with Nonlinear Programming for TwoSided Flight Path Optimization, Ph.D. Thesis, University of Illinois at UrbanaChampaign, Champaign, 2002. 
[12] 
K. Horie and B. A. Conway, Optimal fighter pursuitevasion maneuvers found via twosided optimization, Journal of Guidance, Control, and Dynamics, 29 (2006), 105112. doi: 10.2514/1.3960. 
[13] 
R. Isaacs, Differential Games, John Wiley and Sons, New York, 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, The University of Western Australia, 2002. 
[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), 3053. doi: 10.1007/s1095701200069. 
[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), 162177. 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), 866875. 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), 260291. doi: 10.1007/s1095701199045. 
[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), 275309. 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, Control, and Optimization, 2 (2012), 571599. 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), 22502257. 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. 
[23] 
M. Pontani and B. A. Conway, Optimal interception of evasive missile warheads: Numerical solution of the differential game, Journal of Guidance, Control, and Dynamics, 31 (2008), 11111122. 
[24] 
M. Pontani and B. A. Conway, Numerical solution of the threedimensional orbital pursuitevasion game, Journal of Guidance, Control, and Dynamics, 32 (2009), 474487. 
[25] 
K. Schittkowski, NLPQL: A FORTRAN subroutine for solving constrained nonlinear programming problems, Annals of Operations Research, 5 (1986), 485500. doi: 10.1007/BF02739235. 
[26] 
T. Shima and J. Shinar, Timevarying linear pursuitevasion game models with bounded controls, Journal of Optimization Theory and Applications, 25 (2002), 607618. 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, Navigation, and Control Conference, (1999), 651661. doi: 10.2514/6.19994065. 
[28] 
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Third Edition, Springer, New York, 2002. doi: 10.1007/9780387217383. 
[29] 
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, Essex, 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), 705718. doi: 10.3934/jimo.2009.5.705. 
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