Article
MSC:
26A33,
35R11,
49J15,
49J20,
49K20,
93C20 | MR 4014590 | Zbl 07144941 | DOI: 10.14736/kyb-2019-2-0337
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
fractional optimal control; cooperative systems;; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives
fractional optimal control; cooperative systems;; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives
Summary:
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.
References:
[1] Agrawal, O. P.: Formulation of Euler-Lagrange equations for fractional variational problems. Math. Anal. Appl. 272 (2002), 368-379. DOI 10.1016/S0022-247X(02)00180-4 | MR 1930721
[2] Agrawal, O P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38 (2004), 323-337. DOI 10.1007/s11071-004-3764-6 | MR 2112177
[3] Agrawal, O. P., Baleanu, D. A.: Hamiltonian formulation and direct numerical scheme for fractional optimal control problems. J. Vibration Control 13 (2007), 9-10, 1269-1281. DOI 10.1177/1077546307077467 | MR 2356715
[4] Al-Refai, M., Luchko, Yu.: Maximum principles for the fractional diffusion equations with the Riemann-Liouville fractional derivative and their applications. Fract. Calc. Appl. Anal. 17 (2014), 2, 483-498. DOI 10.2478/s13540-014-0181-5 | MR 3181067
[5] Al-Refai, M., Luchko, Yu.: Maximum principle for the multi-term time fractional diffusion equations with the Riemann-Liouville fractional derivatives. Appl. Math. Comput. 257 (2015), 40-51. DOI 10.1016/j.amc.2014.12.127 | MR 3320647
[6] Al-Refai, M., Luchko, Yu.: Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications. Analysis 36 (2016), 123-133. DOI 10.1515/anly-2015-5011 | MR 3491861
[7] Bahaa, G. M.: Fractional optimal control problem for variational inequalities with control constraints. IMA J. Math. Control Inform. 35 (2018), 1, 107-122. DOI 10.1186/s13662-016-0976-2 | MR 3802084
[8] Bahaa, G. M.: Fractional optimal control problem for differential system with control constraints. Filomat J. 30 (2016), 8, 2177-2189. MR 3583154
[9] Bahaa, G. M.: Fractional optimal control problem for differential system with control constraints with delay argument. Advances Difference Equations 2017 (2017), 69, 1-19. DOI 10.1186/s13662-017-1121-6 | MR 3615527
[10] Bahaa, G. M.: Optimal control for cooperative parabolic systems governed by Schrödinger operator with control constraints. IMA J. Math. Control Inform. 24 (2007), 1-12. DOI 10.1093/imamci/dnl001 | MR 2310985
[11] Bahaa, G. M., Hamiaz, A.: Optimality conditions for fractional differential inclusions with non-singular Mittag-Leffler Kernel. Adv. Difference Equations (2018), 257. MR 3833842
[12] Baleanu, D. A., Agrawal, O. P.: Fractional Hamilton formalism within Caputo's derivative. Czechosl. J. Phys. 56 (2006), 10/11 1087-1092. DOI 10.1007/s10582-006-0406-x | MR 2282282
[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. MR 2800417
[14] Bastos, N. R. O., Ferreira, R. A. C., Torres, D. F. M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Cont. Dyn. Syst. 29 (2011), 2, 417-437. DOI 10.3934/dcds.2011.29.417 | MR 2728463 | Zbl 1209.49020
[15] Eidelman, S. D., Kochubei, A. N.: Cauchy problem for fractional diffusion equations. J. Diff. Equat. 199(2004), 211-255. DOI 10.1016/j.jde.2003.12.002 | MR 2047909
[16] El-Sayed, A. M. A.: Fractional differential equations. Kyungpook Math. J. 28 (1988), 2, 18-22. MR 1053036
[17] Fleckinger, J.: Estimates of the number of eigenvalues for an operator of Schrödinger type. Proc. Royal Soc. Edinburg 89A (1981), 355-361. DOI 10.1017/s0308210500020357 | MR 0635770
[18] Fleckinger, J.: Method of sub-super solutions for some elliptic systems defined on $\Omega$. 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. MR 1296110
[20] Fleckinger, J., Serag, H.: Semilinear cooperative elliptic systems on $R^{n}$. Rend. di Mat. 15 (1995), VII, 89-108. MR 1330181
[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. MR 2386201
[22] Kochubei, A. N.: Fractional order diffusion. Diff. Equations 26 (1990), 485-492. MR 1061448
[23] Kochubei, A. N.: General fractional calculus, evolution equations, and renewal processes. Integr. Equat. Oper. Theory 71 (2011), 583-600. DOI 10.1007/s00020-011-1918-8 | MR 2854867
[24] Kotarski, W., El-Saify, H. A., Bahaa, G. M.: Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay. IMA J. Math. Control Inform. 19 (2002), 4, 461-476. DOI 10.1093/imamci/19.4.461 | MR 1949014
[25] Lions, J. L.: Optimal Control Of Systems Governed By Partial Differential Equations. Springer-Verlag, Band 170 (1971). MR 0271512
[26] Lions, J. L., Magenes, E.: Non-Homogeneous Boundary Value Problem and Applications. Springer-Verlag, New York 1972. DOI 10.1007/978-3-642-65217-2 | MR 0350177
[27] Liu, Y., Rundell, W., Yamamoto, M.: Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fract. Calc. Appl. Anal. 19 (2016), 4, 888-906. DOI 10.1515/fca-2016-0048 | MR 3543685
[28] Liu, Z., Zeng, Sh., Bai, Y.: Maximum principles for multi-term spacetime variable order fractional diffusion equations and their applications. Fract. Calc. Appl. Anal. 19 (2016), 1, 188-211. DOI 10.1515/fca-2016-0011 | MR 3475416
[29] Luchko, Yu.: Maximum principle for the generalized time fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218-223. DOI 10.1016/j.jmaa.2008.10.018 | MR 2472935
[30] Luchko, Yu.: Boundary value problems for the generalized time fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12 (2009), 409-422. MR 2598188
[31] Luchko, Yu.: Some uniqueness and existence results for the initial boundary value problems for the generalized time fractional diffusion equation. Comput. Math. Appl. 59 (2010), 1766-1772. DOI 10.1016/j.camwa.2009.08.015 | MR 2595950
[32] Luchko, Yu.: Initial boundary value problems for the generalized multiterm time fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538-548. DOI 10.1016/j.jmaa.2010.08.048 | MR 2729240
[33] Luchko, Yu., Yamamoto, M.: General time fractional diffusion equation: Some uniqueness and existence results for the initial boundary value problems. Fract. Calc. Appl. Anal. 19 (2016), 3, 676-695. DOI 10.1515/fca-2016-0036 | MR 3563605
[34] Matychyna, I., Onyshchenkob, V.: On time-optimal control of fractional-order systems. J. Comput. Appl. Math. 339 (2018), 245-257. DOI 10.1016/j.cam.2017.10.016 | MR 3787691
[35] Mophou, G. M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61 (2011), 68-78. DOI 10.1016/j.camwa.2010.10.030 | MR 2739436
[36] Mophou, G. M.: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62 (2011), 1413-1426. DOI 10.1016/j.camwa.2011.04.044 | MR 2824729
[37] Oldham, K. B., Spanier, J.: The Fractional Calculus. Academic Press, New York 1974. MR 0361633 | Zbl 0292.26011
[38] Protter, M., Weinberger, H.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Clifs, 1967. DOI 10.1007/978-1-4612-5282-5 | MR 0219861
[39] Ye, H., Liu, F., Anh, V., Turner, I.: Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional differential equations. Appl. Math. Comput. 227 (2014), 531-540. DOI 10.1016/j.amc.2013.11.015 | MR 3146339
Search
Partner of
