Optimal control problem and maximum principle for fractional order cooperative systems

G. M. Bahaa

Kybernetika (2019)

  • Volume: 55, Issue: 2, page 337-358
  • ISSN: 0023-5954

Abstract

top
In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.

How to cite

top

Bahaa, G. M.. "Optimal control problem and maximum principle for fractional order cooperative systems." Kybernetika 55.2 (2019): 337-358. <http://eudml.org/doc/294568>.

@article{Bahaa2019,
abstract = {In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.},
author = {Bahaa, G. M.},
journal = {Kybernetika},
keywords = {fractional optimal control; cooperative systems; ; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives},
language = {eng},
number = {2},
pages = {337-358},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimal control problem and maximum principle for fractional order cooperative systems},
url = {http://eudml.org/doc/294568},
volume = {55},
year = {2019},
}

TY - JOUR
AU - Bahaa, G. M.
TI - Optimal control problem and maximum principle for fractional order cooperative systems
JO - Kybernetika
PY - 2019
PB - Institute of Information Theory and Automation AS CR
VL - 55
IS - 2
SP - 337
EP - 358
AB - In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.
LA - eng
KW - fractional optimal control; cooperative systems; ; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives
UR - http://eudml.org/doc/294568
ER -

References

top
  1. Agrawal, O. P., 10.1016/S0022-247X(02)00180-4, Math. Anal. Appl. 272 (2002), 368-379. MR1930721DOI10.1016/S0022-247X(02)00180-4
  2. Agrawal, O P., 10.1007/s11071-004-3764-6, Nonlinear Dynamics 38 (2004), 323-337. MR2112177DOI10.1007/s11071-004-3764-6
  3. Agrawal, O. P., Baleanu, D. A., 10.1177/1077546307077467, J. Vibration Control 13 (2007), 9-10, 1269-1281. MR2356715DOI10.1177/1077546307077467
  4. Al-Refai, M., Luchko, Yu., 10.2478/s13540-014-0181-5, Fract. Calc. Appl. Anal. 17 (2014), 2, 483-498. MR3181067DOI10.2478/s13540-014-0181-5
  5. Al-Refai, M., Luchko, Yu., 10.1016/j.amc.2014.12.127, Appl. Math. Comput. 257 (2015), 40-51. MR3320647DOI10.1016/j.amc.2014.12.127
  6. Al-Refai, M., Luchko, Yu., 10.1515/anly-2015-5011, Analysis 36 (2016), 123-133. MR3491861DOI10.1515/anly-2015-5011
  7. Bahaa, G. M., 10.1186/s13662-016-0976-2, IMA J. Math. Control Inform. 35 (2018), 1, 107-122. MR3802084DOI10.1186/s13662-016-0976-2
  8. Bahaa, G. M., Fractional optimal control problem for differential system with control constraints., Filomat J. 30 (2016), 8, 2177-2189. MR3583154
  9. Bahaa, G. M., 10.1186/s13662-017-1121-6, Advances Difference Equations 2017 (2017), 69, 1-19. MR3615527DOI10.1186/s13662-017-1121-6
  10. Bahaa, G. M., 10.1093/imamci/dnl001, IMA J. Math. Control Inform. 24 (2007), 1-12. MR2310985DOI10.1093/imamci/dnl001
  11. Bahaa, G. M., Hamiaz, A., Optimality conditions for fractional differential inclusions with non-singular Mittag-Leffler Kernel., Adv. Difference Equations (2018), 257. MR3833842
  12. Baleanu, D. A., Agrawal, O. P., 10.1007/s10582-006-0406-x, Czechosl. J. Phys. 56 (2006), 10/11 1087-1092. MR2282282DOI10.1007/s10582-006-0406-x
  13. Bastos, N. R. O., Mozyrska, D., Torres, D. F. M., Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform., Int. J. Math. Comput. 11 (2011), J11, 1-9. MR2800417
  14. Bastos, N. R. O., Ferreira, R. A. C., Torres, D. F. M., 10.3934/dcds.2011.29.417, Discrete Cont. Dyn. Syst. 29 (2011), 2, 417-437. Zbl1209.49020MR2728463DOI10.3934/dcds.2011.29.417
  15. Eidelman, S. D., Kochubei, A. N., 10.1016/j.jde.2003.12.002, J. Diff. Equat. 199(2004), 211-255. MR2047909DOI10.1016/j.jde.2003.12.002
  16. El-Sayed, A. M. A., Fractional differential equations., Kyungpook Math. J. 28 (1988), 2, 18-22. MR1053036
  17. Fleckinger, J., 10.1017/s0308210500020357, Proc. Royal Soc. Edinburg 89A (1981), 355-361. MR0635770DOI10.1017/s0308210500020357
  18. Fleckinger, J., Method of sub-super solutions for some elliptic systems defined on Ω ., Preprint UMR MIP, Universite Toulouse 3 (1994). 
  19. Fleckinger, J., Hernándes, J., Thélin, F. de, On maximum principle and existence of positive solutions for cooperative elliptic systems., Diff. Int. Eqns. 8 (1995), 69-85. MR1296110
  20. Fleckinger, J., Serag, H., Semilinear cooperative elliptic systems on R n ., Rend. di Mat. 15 (1995), VII, 89-108. MR1330181
  21. Gastao, S. F. Frederico, Torres, D. F. M., Fractional optimal control in the sense of Caputo and the fractional Noether's theorem., Int. Math. Forum 3 (2008), 10, 479-493. MR2386201
  22. Kochubei, A. N., Fractional order diffusion., Diff. Equations 26 (1990), 485-492. MR1061448
  23. Kochubei, A. N., 10.1007/s00020-011-1918-8, Integr. Equat. Oper. Theory 71 (2011), 583-600. MR2854867DOI10.1007/s00020-011-1918-8
  24. Kotarski, W., El-Saify, H. A., Bahaa, G. M., 10.1093/imamci/19.4.461, IMA J. Math. Control Inform. 19 (2002), 4, 461-476. MR1949014DOI10.1093/imamci/19.4.461
  25. Lions, J. L., Optimal Control Of Systems Governed By Partial Differential Equations., Springer-Verlag, Band 170 (1971). MR0271512
  26. Lions, J. L., Magenes, E., 10.1007/978-3-642-65217-2, Springer-Verlag, New York 1972. MR0350177DOI10.1007/978-3-642-65217-2
  27. Liu, Y., Rundell, W., Yamamoto, M., 10.1515/fca-2016-0048, Fract. Calc. Appl. Anal. 19 (2016), 4, 888-906. MR3543685DOI10.1515/fca-2016-0048
  28. Liu, Z., Zeng, Sh., Bai, Y., 10.1515/fca-2016-0011, Fract. Calc. Appl. Anal. 19 (2016), 1, 188-211. MR3475416DOI10.1515/fca-2016-0011
  29. Luchko, Yu., 10.1016/j.jmaa.2008.10.018, J. Math. Anal. Appl. 351 (2009), 218-223. MR2472935DOI10.1016/j.jmaa.2008.10.018
  30. Luchko, Yu., Boundary value problems for the generalized time fractional diffusion equation of distributed order., Fract. Calc. Appl. Anal. 12 (2009), 409-422. MR2598188
  31. Luchko, Yu., 10.1016/j.camwa.2009.08.015, Comput. Math. Appl. 59 (2010), 1766-1772. MR2595950DOI10.1016/j.camwa.2009.08.015
  32. Luchko, Yu., 10.1016/j.jmaa.2010.08.048, J. Math. Anal. Appl. 374 (2011), 538-548. MR2729240DOI10.1016/j.jmaa.2010.08.048
  33. Luchko, Yu., Yamamoto, M., 10.1515/fca-2016-0036, Fract. Calc. Appl. Anal. 19 (2016), 3, 676-695. MR3563605DOI10.1515/fca-2016-0036
  34. Matychyna, I., Onyshchenkob, V., 10.1016/j.cam.2017.10.016, J. Comput. Appl. Math. 339 (2018), 245-257. MR3787691DOI10.1016/j.cam.2017.10.016
  35. Mophou, G. M., 10.1016/j.camwa.2010.10.030, Comput. Math. Appl. 61 (2011), 68-78. MR2739436DOI10.1016/j.camwa.2010.10.030
  36. Mophou, G. M., 10.1016/j.camwa.2011.04.044, Comput. Math. Appl. 62 (2011), 1413-1426. MR2824729DOI10.1016/j.camwa.2011.04.044
  37. Oldham, K. B., Spanier, J., The Fractional Calculus., Academic Press, New York 1974. Zbl0292.26011MR0361633
  38. Protter, M., Weinberger, H., 10.1007/978-1-4612-5282-5, Prentice Hall, Englewood Clifs, 1967. MR0219861DOI10.1007/978-1-4612-5282-5
  39. Ye, H., Liu, F., Anh, V., Turner, I., 10.1016/j.amc.2013.11.015, Appl. Math. Comput. 227 (2014), 531-540. MR3146339DOI10.1016/j.amc.2013.11.015

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.