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Error bounds for Euler approximation of linearquadratic control problems with bangbang solutions
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results
1.  Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845 
2.  Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, Western Australia 6845, Australia 
References:
[1] 
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with nonconvex control constraints, Automatica, 47 (2011), 341347. doi: 10.1016/j.automatica.2010.10.037. 
[2] 
N. U. Ahmed, "Dynamic Systems and Control with Applications," World Scientific, Singapore, 2006. doi: 10.1142/6262. 
[3] 
C. Büskens and H. Maurer, SQPmethods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and realtime control, Journal of Computational and Applied Mathematics, 120 (2000), 85108. doi: 10.1016/S03770427(00)003058. 
[4] 
M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247270. 
[5] 
C. J. Goh and K. L. Teo, Control parametrization: A unified approach to optimal control problems with general constraints, Automatica, 24 (1988), 318. doi: 10.1016/00051098(88)900039. 
[6] 
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371375. doi: 10.1016/00051098(90)90131Z. 
[7] 
A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover edition, Dover Publications, New York, 1975. 
[8] 
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. 
[9] 
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7 (2011), 6381. 
[10] 
R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 29232929. doi: 10.1016/j.automatica.2008.04.011. 
[11] 
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switchedcapacitor DC/DC power converter, Automatica, 45 (2009), 973980. doi: 10.1016/j.automatica.2008.10.031. 
[12] 
R. 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. 
[13] 
D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, Springer, New York, 2008. 
[14] 
J. Nocedal and S. J. Wright, "Numerical Optimization," 2nd edition, Springer, New York, 2006. 
[15] 
H. L. Royden and P. M. Fitzpatrick, "Real Analysis," 4th edition, Prentice Hall, Boston, 2010. 
[16] 
W. Rudin, "Principles of Mathematical Analysis," 3rd edition, McGrawHill, New York, 1976. 
[17] 
K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Longman Scientific and Technical, Essex, 1991. 
[18] 
K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control, Journal of Optimization Theory and Applications, 68 (1991), 335357. doi: 10.1007/BF00941572. 
[19] 
K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica, 29 (1993), 789792. doi: 10.1016/00051098(93)900766. 
[20] 
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. 
[21] 
C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty method for semiinfinite programming problems, Journal of Industrial and Management Optimization, 6 (2010), 895910. doi: 10.3934/jimo.2010.6.895. 
[22] 
Y. Zhao and M. A. Stadtherr, Rigorous global optimization method for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 1267812693. doi: http://dx.doi.org/10.1021/ie200996f. 
show all references
References:
[1] 
B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with nonconvex control constraints, Automatica, 47 (2011), 341347. doi: 10.1016/j.automatica.2010.10.037. 
[2] 
N. U. Ahmed, "Dynamic Systems and Control with Applications," World Scientific, Singapore, 2006. doi: 10.1142/6262. 
[3] 
C. Büskens and H. Maurer, SQPmethods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and realtime control, Journal of Computational and Applied Mathematics, 120 (2000), 85108. doi: 10.1016/S03770427(00)003058. 
[4] 
M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247270. 
[5] 
C. J. Goh and K. L. Teo, Control parametrization: A unified approach to optimal control problems with general constraints, Automatica, 24 (1988), 318. doi: 10.1016/00051098(88)900039. 
[6] 
L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371375. doi: 10.1016/00051098(90)90131Z. 
[7] 
A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover edition, Dover Publications, New York, 1975. 
[8] 
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. 
[9] 
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems, Pacific Journal of Optimization, 7 (2011), 6381. 
[10] 
R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 29232929. doi: 10.1016/j.automatica.2008.04.011. 
[11] 
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling, Optimal switching instants for a switchedcapacitor DC/DC power converter, Automatica, 45 (2009), 973980. doi: 10.1016/j.automatica.2008.10.031. 
[12] 
R. 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. 
[13] 
D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, Springer, New York, 2008. 
[14] 
J. Nocedal and S. J. Wright, "Numerical Optimization," 2nd edition, Springer, New York, 2006. 
[15] 
H. L. Royden and P. M. Fitzpatrick, "Real Analysis," 4th edition, Prentice Hall, Boston, 2010. 
[16] 
W. Rudin, "Principles of Mathematical Analysis," 3rd edition, McGrawHill, New York, 1976. 
[17] 
K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems," Longman Scientific and Technical, Essex, 1991. 
[18] 
K. L. Teo and L. S. Jennings, Optimal control with a cost on changing control, Journal of Optimization Theory and Applications, 68 (1991), 335357. doi: 10.1007/BF00941572. 
[19] 
K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica, 29 (1993), 789792. doi: 10.1016/00051098(93)900766. 
[20] 
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. 
[21] 
C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty method for semiinfinite programming problems, Journal of Industrial and Management Optimization, 6 (2010), 895910. doi: 10.3934/jimo.2010.6.895. 
[22] 
Y. Zhao and M. A. Stadtherr, Rigorous global optimization method for dynamic systems subject to inequality path constraints, Industrial and Engineering Chemistry Research, 50 (2011), 1267812693. doi: http://dx.doi.org/10.1021/ie200996f. 
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