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
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show all references

References:
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Automatica, 47 (2011), 341-347. doi: 10.1016/j.automatica.2010.10.037.  Google Scholar

[2]

World Scientific, Singapore, 2006. doi: 10.1142/6262.  Google Scholar

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Journal of Computational and Applied Mathematics, 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[4]

Journal of Industrial and Management Optimization, 4 (2008), 247-270.  Google Scholar

[5]

Automatica, 24 (1988), 3-18. doi: 10.1016/0005-1098(88)90003-9.  Google Scholar

[6]

Automatica, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

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Dover edition, Dover Publications, New York, 1975.  Google Scholar

[8]

Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.  Google Scholar

[9]

Pacific Journal of Optimization, 7 (2011), 63-81.  Google Scholar

[10]

Automatica, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.  Google Scholar

[11]

Automatica, 45 (2009), 973-980. doi: 10.1016/j.automatica.2008.10.031.  Google Scholar

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[13]

3rd edition, Springer, New York, 2008.  Google Scholar

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2nd edition, Springer, New York, 2006.  Google Scholar

[15]

4th edition, Prentice Hall, Boston, 2010. Google Scholar

[16]

3rd edition, McGraw-Hill, New York, 1976.  Google Scholar

[17]

Longman Scientific and Technical, Essex, 1991.  Google Scholar

[18]

Journal of Optimization Theory and Applications, 68 (1991), 335-357. doi: 10.1007/BF00941572.  Google Scholar

[19]

Automatica, 29 (1993), 789-792. doi: 10.1016/0005-1098(93)90076-6.  Google Scholar

[20]

Journal of Industrial and Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705.  Google Scholar

[21]

Journal of Industrial and Management Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895.  Google Scholar

[22]

Industrial and Engineering Chemistry Research, 50 (2011), 12678-12693. doi: http://dx.doi.org/10.1021/ie200996f.  Google Scholar

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