# American Institute of Mathematical Sciences

2012, 2(3): 571-599. doi: 10.3934/naco.2012.2.571

## 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

Received  March 2012 Revised  May 2012 Published  August 2012

Control parameterization is a powerful numerical technique for solving optimal control problems with general nonlinear constraints. The main idea of control parameterization is to discretize the control space by approximating the control by a piecewise-constant or piecewise-linear function, thereby yielding an approximate nonlinear programming problem. This approximate problem can then be solved using standard gradient-based optimization techniques. In this paper, we consider the control parameterization method for a class of optimal control problems in which the admissible controls are functions of bounded variation and the state and control are subject to continuous inequality constraints. We show that control parameterization generates a sequence of suboptimal controls whose costs converge to the true optimal cost. This result has previously only been proved for the case when the admissible controls are restricted to piecewise continuous functions.
Citation: Ryan Loxton, Qun Lin, Volker Rehbock, Kok Lay Teo. Control parameterization for optimal control problems with continuous inequality constraints: New convergence results. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 571-599. doi: 10.3934/naco.2012.2.571
##### References:
 [1] B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints,, Automatica, 47 (2011), 341. doi: 10.1016/j.automatica.2010.10.037. [2] N. U. Ahmed, "Dynamic Systems and Control with Applications,", World Scientific, (2006). doi: 10.1142/6262. [3] 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,, Journal of Computational and Applied Mathematics, 120 (2000), 85. doi: 10.1016/S0377-0427(00)00305-8. [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), 247. [5] C. J. Goh and K. L. Teo, Control parametrization: A unified approach to optimal control problems with general constraints,, Automatica, 24 (1988), 3. doi: 10.1016/0005-1098(88)90003-9. [6] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. [7] A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Dover edition, (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), 260. doi: 10.1007/s10957-011-9904-5. [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), 63. [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), 2923. 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 switched-capacitor DC/DC power converter,, Automatica, 45 (2009), 973. 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), 2250. doi: 10.1016/j.automatica.2009.05.029. [13] D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, (2008). [14] J. Nocedal and S. J. Wright, "Numerical Optimization,", 2nd edition, (2006). [15] H. L. Royden and P. M. Fitzpatrick, "Real Analysis,", 4th edition, (2010). [16] W. Rudin, "Principles of Mathematical Analysis,", 3rd edition, (1976). [17] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems,", Longman Scientific and Technical, (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), 335. 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), 789. doi: 10.1016/0005-1098(93)90076-6. [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), 705. doi: 10.3934/jimo.2009.5.705. [21] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty method for semi-infinite programming problems,, Journal of Industrial and Management Optimization, 6 (2010), 895. 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), 12678. doi: http://dx.doi.org/10.1021/ie200996f.

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##### References:
 [1] B. Açikmeşe and L. Blackmore, Lossless convexification of a class of optimal control problems with non-convex control constraints,, Automatica, 47 (2011), 341. doi: 10.1016/j.automatica.2010.10.037. [2] N. U. Ahmed, "Dynamic Systems and Control with Applications,", World Scientific, (2006). doi: 10.1142/6262. [3] 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,, Journal of Computational and Applied Mathematics, 120 (2000), 85. doi: 10.1016/S0377-0427(00)00305-8. [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), 247. [5] C. J. Goh and K. L. Teo, Control parametrization: A unified approach to optimal control problems with general constraints,, Automatica, 24 (1988), 3. doi: 10.1016/0005-1098(88)90003-9. [6] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. [7] A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Dover edition, (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), 260. doi: 10.1007/s10957-011-9904-5. [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), 63. [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), 2923. 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 switched-capacitor DC/DC power converter,, Automatica, 45 (2009), 973. 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), 2250. doi: 10.1016/j.automatica.2009.05.029. [13] D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, (2008). [14] J. Nocedal and S. J. Wright, "Numerical Optimization,", 2nd edition, (2006). [15] H. L. Royden and P. M. Fitzpatrick, "Real Analysis,", 4th edition, (2010). [16] W. Rudin, "Principles of Mathematical Analysis,", 3rd edition, (1976). [17] K. L. Teo, C. J. Goh and K. H. Wong, "A Unified Computational Approach to Optimal Control Problems,", Longman Scientific and Technical, (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), 335. 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), 789. doi: 10.1016/0005-1098(93)90076-6. [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), 705. doi: 10.3934/jimo.2009.5.705. [21] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty method for semi-infinite programming problems,, Journal of Industrial and Management Optimization, 6 (2010), 895. 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), 12678. doi: http://dx.doi.org/10.1021/ie200996f.
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