Estimation of the Optimal Regularization Parameters in Optimal Control Problems with time delay
DOI:
https://doi.org/10.24297/jam.v12i9.129Keywords:
Differential Operator, Tikhonov Regularization, Regularization parameter, L-curve method, discrepancy principle.Abstract
In this paper we use the L-curve method and the Morozov discrepancy principle for the estimation of the regularization parameter in the regularization of time-delayed optimal control computation. Zeroth order, first order and second order differential operators are considered. Two test examples show that the L-curve method and the two discrepancy principles give close estimations for the regularization parameters.Downloads
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References
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[2] Gölmann, L., Kern, D., and Maurer, H. (2009). Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Applications and Methods, 30(4):341–365.
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[10] Wong, K. H. (1988). A control parametrization algorithm for nonlinear time-lag optimal control problems. Opsearch, 25(E):177–184.
[11] Wong, K. H., Jennings, L., and Benyah, F. (2001). Control parametrization method for free planning time optimal control problems with time-delayed arguments. Journal of Nonlinear Analysis, 47:5679–5689.
[12] Wong, K. H., Jennings, L., and Benyah, F. (2002). The control parametrization enhancing transform for constrained time-delayed optimal control problems. ANZIAM J. The Australian and New Zealand Industrial and Applied Mathematics Journal, 43(E):E154–E185.
[13] Betts, J. T. and Hoffman, W. (1999). Exploring sparsity in the direct transcription method for optimal control. Computational Optimization and Applications, 14:179–201.
[14] Coleman, T., Branch, M., and Grace, A. (1999). Optimization Toolbox for Use With Matlab. The Mathworks Inc.
[15] Boggs, P. T. and Tolle, J. W. (1996). Sequential quadratic programming. Acta Numerica, pages 1–48.
[16] Lawrence, C. T. and Tits, A. L. (2001). A computationally efficient feasible sequential quadratic programming algorithm. SIAM Journal on Optimization, 11:1092–1118.
[17] Matveev, A. (2005). The instability of optimal control problems to time delay. SIAM J. Control Optimization, 43(5):1757–1786.
[18] R. Ramlau. Morozov's discrepancy principle for tikhonov regularization of nonlinear operators. Technical Report 01-08, Zentrum fur Technomathematik, University of Bremen, 2001.
[19] Benyah, F and Jennings L.S (1998), The l-curve in regularization of optimal control computations. Journal of Australian Mathematical Society, 40(E):E138--E172.
[20] F. Benyah and L. S. Jennings. A review of ill-conditioning and regularization in optimal control computation. In Eds X., Yang, K. L. Teo, and L. Caccetta, editors, Optimization Methods and Applications, pages 23{44. Kluwer Academic publishers, Dordrecht, The Netherlands, 2001.
[2] Gölmann, L., Kern, D., and Maurer, H. (2009). Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Applications and Methods, 30(4):341–365.
[3] Lin, J. (1996). Optimal control of time-delay systems by forward iterative dynamic programming. Ind. Eng. Chem. Res., 35(8):2795–2800.
[4] Luus, R. (2000). Iterative Dynamic Programming. Charman and Hall/CRC.
[5] Williams, B. (2007). Optimal management of non-Markovian biological population. Ecological Modelling, 200:234–242.
[6] Betts, J.T. (2001). Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics.
[7] Shwartz, A. (1996). Theory and Implementation of Numerical Methods Based on Runge-Kutta Integration for Solving Optimal Control Problems. Phd thesis, Electronic Research Laboratory, UC Burkeley.
[8] Inanc, T. and Bhattacharya, R. (2003). Numerical solution of optimal control problems using splines.
[9] Von Stryk, O. (1993). Numerical solution of optimal control problems by direct collocation. International Series of Numerical Mathematics, 111:129–143.
[10] Wong, K. H. (1988). A control parametrization algorithm for nonlinear time-lag optimal control problems. Opsearch, 25(E):177–184.
[11] Wong, K. H., Jennings, L., and Benyah, F. (2001). Control parametrization method for free planning time optimal control problems with time-delayed arguments. Journal of Nonlinear Analysis, 47:5679–5689.
[12] Wong, K. H., Jennings, L., and Benyah, F. (2002). The control parametrization enhancing transform for constrained time-delayed optimal control problems. ANZIAM J. The Australian and New Zealand Industrial and Applied Mathematics Journal, 43(E):E154–E185.
[13] Betts, J. T. and Hoffman, W. (1999). Exploring sparsity in the direct transcription method for optimal control. Computational Optimization and Applications, 14:179–201.
[14] Coleman, T., Branch, M., and Grace, A. (1999). Optimization Toolbox for Use With Matlab. The Mathworks Inc.
[15] Boggs, P. T. and Tolle, J. W. (1996). Sequential quadratic programming. Acta Numerica, pages 1–48.
[16] Lawrence, C. T. and Tits, A. L. (2001). A computationally efficient feasible sequential quadratic programming algorithm. SIAM Journal on Optimization, 11:1092–1118.
[17] Matveev, A. (2005). The instability of optimal control problems to time delay. SIAM J. Control Optimization, 43(5):1757–1786.
[18] R. Ramlau. Morozov's discrepancy principle for tikhonov regularization of nonlinear operators. Technical Report 01-08, Zentrum fur Technomathematik, University of Bremen, 2001.
[19] Benyah, F and Jennings L.S (1998), The l-curve in regularization of optimal control computations. Journal of Australian Mathematical Society, 40(E):E138--E172.
[20] F. Benyah and L. S. Jennings. A review of ill-conditioning and regularization in optimal control computation. In Eds X., Yang, K. L. Teo, and L. Caccetta, editors, Optimization Methods and Applications, pages 23{44. Kluwer Academic publishers, Dordrecht, The Netherlands, 2001.
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Published
2016-09-21
How to Cite
Bashier, E. B. M. (2016). Estimation of the Optimal Regularization Parameters in Optimal Control Problems with time delay. JOURNAL OF ADVANCES IN MATHEMATICS, 12(9), 6589–6602. https://doi.org/10.24297/jam.v12i9.129
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