An efficient hp spectral collocation method for nonsmooth optimal control problems

Mehrnoosh Hedayati; Hojjat Ahsani Tehrani; Alireza Fakharzadeh Jahromi; Mohammad Hadi Noori Skandari; Dumitru Baleanu

Kybernetika (2022)

  • Volume: 58, Issue: 6, page 843-862
  • ISSN: 0023-5954

Abstract

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One of the most challenging problems in the optimal control theory consists of solving the nonsmooth optimal control problems where several discontinuities may be present in the control variable and derivative of the state variable. Recently some extended spectral collocation methods have been introduced for solving such problems, and a matrix of differentiation is usually used to discretize and to approximate the derivative of the state variable in the particular collocation points. In such methods, there is typically no condition for the continuity of the state variable at the switching points. In this article, we propose an efficient hp spectral collocation method for the general form of nonsmooth optimal control problems based on the operational integration matrix. The time interval of the problem is first partitioned into several variable subintervals, and the problem is then discretized by considering the Legendre-Gauss-Lobatto collocation points. Here, the switching points are unknown parameters, and having solved the final discretized problem, we achieve some approximations for the optimal solutions and the switching points. We solve some comparative numerical test problems to support of the performance of the suggested approach.

How to cite

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Hedayati, Mehrnoosh, et al. "An efficient hp spectral collocation method for nonsmooth optimal control problems." Kybernetika 58.6 (2022): 843-862. <http://eudml.org/doc/299519>.

@article{Hedayati2022,
abstract = {One of the most challenging problems in the optimal control theory consists of solving the nonsmooth optimal control problems where several discontinuities may be present in the control variable and derivative of the state variable. Recently some extended spectral collocation methods have been introduced for solving such problems, and a matrix of differentiation is usually used to discretize and to approximate the derivative of the state variable in the particular collocation points. In such methods, there is typically no condition for the continuity of the state variable at the switching points. In this article, we propose an efficient hp spectral collocation method for the general form of nonsmooth optimal control problems based on the operational integration matrix. The time interval of the problem is first partitioned into several variable subintervals, and the problem is then discretized by considering the Legendre-Gauss-Lobatto collocation points. Here, the switching points are unknown parameters, and having solved the final discretized problem, we achieve some approximations for the optimal solutions and the switching points. We solve some comparative numerical test problems to support of the performance of the suggested approach.},
author = {Hedayati, Mehrnoosh, Ahsani Tehrani, Hojjat, Fakharzadeh Jahromi, Alireza, Noori Skandari, Mohammad Hadi, Baleanu, Dumitru},
journal = {Kybernetika},
keywords = {nonsmooth optimal control; hp-method; Lagrange interpolating polynomials; Legendre-Gauss-Lobatto points},
language = {eng},
number = {6},
pages = {843-862},
publisher = {Institute of Information Theory and Automation AS CR},
title = {An efficient hp spectral collocation method for nonsmooth optimal control problems},
url = {http://eudml.org/doc/299519},
volume = {58},
year = {2022},
}

TY - JOUR
AU - Hedayati, Mehrnoosh
AU - Ahsani Tehrani, Hojjat
AU - Fakharzadeh Jahromi, Alireza
AU - Noori Skandari, Mohammad Hadi
AU - Baleanu, Dumitru
TI - An efficient hp spectral collocation method for nonsmooth optimal control problems
JO - Kybernetika
PY - 2022
PB - Institute of Information Theory and Automation AS CR
VL - 58
IS - 6
SP - 843
EP - 862
AB - One of the most challenging problems in the optimal control theory consists of solving the nonsmooth optimal control problems where several discontinuities may be present in the control variable and derivative of the state variable. Recently some extended spectral collocation methods have been introduced for solving such problems, and a matrix of differentiation is usually used to discretize and to approximate the derivative of the state variable in the particular collocation points. In such methods, there is typically no condition for the continuity of the state variable at the switching points. In this article, we propose an efficient hp spectral collocation method for the general form of nonsmooth optimal control problems based on the operational integration matrix. The time interval of the problem is first partitioned into several variable subintervals, and the problem is then discretized by considering the Legendre-Gauss-Lobatto collocation points. Here, the switching points are unknown parameters, and having solved the final discretized problem, we achieve some approximations for the optimal solutions and the switching points. We solve some comparative numerical test problems to support of the performance of the suggested approach.
LA - eng
KW - nonsmooth optimal control; hp-method; Lagrange interpolating polynomials; Legendre-Gauss-Lobatto points
UR - http://eudml.org/doc/299519
ER -

References

top
  1. Agamawi, Y., Hager, W., Rao, A. V., Mesh refinement method for optimal control problems with discontinuous control profiles., In: AIAA Guidance, Navigation, and Control Conference 2017, pp. 1506. 
  2. Aly, G. M., Chan, W. C., , Int. J. Control 17 (1973), 809-815. DOI
  3. Aronna, M. S., Bonnans, J. F., Martinon, P., , J. Optim. Theory Appl. 158 (2013), 419-459. MR3084385DOI
  4. Betts, J. T., Practical methods for optimal control using nonlinear programming, ser., In: Advances in Design and Control, SIAM Press, Philadelphia 2001, 3. MR1826768
  5. Betts, J. T., Huffman, W. P., 10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q, J. Optimal Control Appl. Methods 19 (1998), 1-21. MR1623173DOI10.1002/(SICI)1099-1514(199801/02)19:1<1::AID-OCA616>3.0.CO;2-Q
  6. Berkmann, P., Pesch, H. J., , J. Optim. Theory Appl.85 (1995), 21-57. MR1330841DOI
  7. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T. A., Spectral Methods in Fluid Dynamics., Springer Series Comput. Physics, Springer, Berlin 1991. MR2340254
  8. Cuthreli, J. E., Biegler, L. T., , J. Amer. Inst. Chemical Engineers 33 (1987), 1257-1270. MR0909947DOI
  9. Cuthrell, J. E., Biegler, L. T., , J. Computers Chemical Engrg. 13 (1989), 49-62. DOI
  10. Dadebo, S. A., McAuley, K. B., On the computation of optimal singular controls., In: Proc. International Conference on Control Applications. IEEE (1995), pp. 150-155. 
  11. Dadebo, S. A., McAuley, K. B., McLellan, P. J., , J. Optim. Control Appl. Methods 19 (1998), 287-297. MR1650209DOI
  12. Darby, C. L., Hager, W., Rao, A. V., , J. Optimal Control Appl. Methods 32 (2011), 476-502. MR2850736DOI
  13. Dolan, E., More, J. J., Munson, T. S., Benchmarking optimization software with COPS 3.0, ANL/ MCS-273., ANL/MCS-TM-273. Argonne National Lab., Argonne, IL (US), 2004. 
  14. Foroozandeh, Z., Shamsi, M., Azhmyakov, V., Shafiee, M., , Math. Methods Appl. Sci. 40 (2016), 1783-1793. MR3622433DOI
  15. Fu, W., Lu, Q., , Complexity (2020), 1-15. DOI
  16. Gao, W., Jiang, Y., Jiang, Z. P., Chai, T., , J. Automatica 72 (2016), 37-45. MR3542912DOI
  17. Goddard, R. H., , Nature 105 (1920), 809-811. DOI
  18. Graichen, K., Petit, N., , IFAC Proceed. 41 (2008), 14301-14306. DOI
  19. Hager, W., Liu, J., Mohapatra, S., Rao, V., Wang, X-SH., , SIAM J. Control Optim. 56 (2017), 1386-1411. MR3784105DOI
  20. Henriques, J. C. C., Lemos, J. M., Gato, L. M C., Falcao, A. F. O., , J. Automatica 85 (2017), 70-82. MR3712847DOI
  21. Hu, G. S., Ong, C. J., Teo, C. L., , J. Optim. Theory Appl. 120 (2004), 395-416. MR2044903DOI
  22. Huang, H. P., McClamroch, N. H., 10.1109/56.2077, IEEE J, Robotics Automat. 4 (1998), 140-149. DOI10.1109/56.2077
  23. Jain, D., Tsiotras, P., , J. Guidance Control Dynamics 31 (2008), 1424-1436. DOI
  24. Kim, J. H. R., Maurer, H., Astrov, Y. A., Bode, M., Purwins, H. G., , J. Comput. Phys. 170 (2001), 395-414. DOI
  25. Ledzewicz, U., Schattler, H., , J. Optim. Theory Appl. 114 (2002), 609-637. MR1921169DOI
  26. Luus, R., , IEEE Trans. Automat. Control 37 (1992), 1802-1806. MR1195226DOI
  27. Martinon, P., Bonnans, F., Laurent-Varin, J., Trelat, E., , J. Guidance Control Dynamics 32 (2009), 51-55. DOI
  28. Marzban, H. R., Hoseini, S. M., , Commun. Nonlinear Sci. Numerical Simul- 18 (2012), 1347-1361. Zbl1282.65075MR3016889DOI
  29. Maurer, H., 10.1007/BF00935706, J. Optim. Theory Appl. 18 (1973), 235-259. MR0408246DOI10.1007/BF00935706
  30. Mehra, R. K., Davis, R. E., , IEEE Trans. Automat. Control 17 (1972), 69-79. MR0368899DOI
  31. Olsder, G. J., , SIAM J. Control Optim. 40 (2001), 1087-1106. MR1882726DOI
  32. Ross, I. M., Gong, Q., Kang, W., , Inst. Electrical and Electronic Engineers Transactions on Automatic Control 51 (2006), 1115-1129. MR2238794DOI
  33. Savku, E., Weber, G. W., , J. Optim. Theory Appl. 179 (2018), 696-721. MR3865345DOI
  34. Seywald, H., Cliff, E. M., , J. Guidance Control Dynamics 16 (1993), 776-781. DOI
  35. Shamsi, M., , J. Optim. Control Appl. Methods 32 (2011), 668-680. MR2871837DOI
  36. Shen, J., Tang, T., Wang, L., Spectral Methods Algorithms, Analysis and Applications., Springer Series in Computational Mathematics 2011. MR2867779
  37. Shirin, A., Klickstein, I. S., Feng, S., Lin, Y. T., Hlavacek, W. S., Sorrentino, F., Prediction of optimal drug schedules for controlling autophagy., Nature 9 (2019), 1-15. 
  38. Signori, A., 10.3233/ASY-191546, Asymptotic Analysis 117 (2020), 4-66. MR4158326DOI10.3233/ASY-191546
  39. Skandari, M. H. N., Tohidi, E., 10.4236/am.2011.25085, Appl. Math. 2 (2011), 646-652. MR2910173DOI10.4236/am.2011.25085
  40. Speyer, J. L., , J. Guidance Control Dynamics 19 (1996), 745-755. DOI
  41. Sun, D. Y., , ISA Trans. 49 (2010), 106-113. DOI
  42. Tabrizidooz, H. R., Marzban, H. R., Pourbabaee, M., Hedayati, M., 10.1016/j.jfranklin.2017.01.002, J. Franklin Inst. 35 (2017), 2393-2414. MR3623593DOI10.1016/j.jfranklin.2017.01.002
  43. Trelat, E., , J. Optim. Theory Appl. 154 (2012), 713-758. MR2957013DOI
  44. Tsiotras, P., Kelley, H. J., , J. Guidance Control Dynamics 15 (1992), 289-296. DOI
  45. Williams, P., , In: Proc. Seventh Biennial Engineering Mathematics and Applications Conference 2005, The Anzim Journal 47 (2006), C101-C115. MR2242566DOI
  46. Zhang, X. Y., 10.1016/j.cam.2017.02.040, J. Computat. Appl. Math. 321 (2017), 284-301. MR3634936DOI10.1016/j.cam.2017.02.040
  47. Zhao, Y., Tsiotras, P., A density-function based mesh refinement algorithm for solving optimal control problems., In: Infotech and Aerospace Conference 2009, 2009-2019. 

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