Kybernetika 55 no. 2, 337-358, 2019

Optimal control problem and maximum principle for fractional order cooperative systems

G. M. BahaaDOI: 10.14736/kyb-2019-2-0337

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.

Keywords:

optimality conditions, fractional optimal control, cooperative systems;, Schrödinger operator, maximum principle, existence of solution, boundary control, fractional Caputo derivatives, Riemann-Liouville derivatives

Classification:

26A33, 49J20, 35R11, 49J15, 49K20, 93C20

References:

  1. O. P. Agrawal: Formulation of Euler-Lagrange equations for fractional variational problems. Math. Anal. Appl. 272 (2002), 368-379.   CrossRef
  2. O P. Agrawal: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics 38 (2004), 323-337.   DOI:10.1007/s11071-004-3764-6
  3. O. P. Agrawal and D. A. Baleanu: Hamiltonian formulation and direct numerical scheme for fractional optimal control problems. J. Vibration Control 13 (2007), 9-10, 1269-1281.   DOI:10.1177/1077546307077467
  4. M. Al-Refai and Yu. Luchko: 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
  5. M. Al-Refai and Yu. Luchko: 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
  6. M. Al-Refai and Yu. Luchko: Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications. Analysis 36 (2016), 123-133.   DOI:10.1515/anly-2015-5011
  7. G. M. Bahaa: 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
  8. G. M. Bahaa: Fractional optimal control problem for differential system with control constraints. Filomat J. 30 (2016), 8, 2177-2189.   CrossRef
  9. G. M. Bahaa: 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
  10. G. M. Bahaa: 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
  11. G. M. Bahaa and A. Hamiaz: Optimality conditions for fractional differential inclusions with non-singular Mittag-Leffler Kernel. Adv. Difference Equations (2018), 257.   CrossRef
  12. D. A. Baleanu and O. P. Agrawal: Fractional Hamilton formalism within Caputo's derivative. Czechosl. J. Phys. 56 (2006), 10/11 1087-1092.   DOI:10.1007/s10582-006-0406-x
  13. N. R. O. Bastos, D. Mozyrska and D. F. M. Torres: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput. 11 (2011), J11, 1-9.   CrossRef
  14. N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: 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
  15. S. D. Eidelman and A. N. Kochubei: Cauchy problem for fractional diffusion equations. J. Diff. Equat. 199(2004), 211-255.   DOI:10.1016/j.jde.2003.12.002
  16. A. M. A. El-Sayed: Fractional differential equations. Kyungpook Math. J. 28 (1988), 2, 18-22.   CrossRef
  17. J. Fleckinger: Estimates of the number of eigenvalues for an operator of Schrödinger type. Proc. Royal Soc. Edinburg 89A (1981), 355-361.   DOI:10.1017/s0308210500020357
  18. J. Fleckinger: Method of sub-super solutions for some elliptic systems defined on $\Omega$. Preprint UMR MIP, Universite Toulouse 3 (1994).   CrossRef
  19. J. Fleckinger, J. Hernándes and F. de Thélin: On maximum principle and existence of positive solutions for cooperative elliptic systems. Diff. Int. Eqns. 8 (1995), 69-85.   CrossRef
  20. J. Fleckinger and H. Serag: Semilinear cooperative elliptic systems on $R^{n}$. Rend. di Mat. 15 (1995), VII, 89-108.   CrossRef
  21. S. F. Frederico Gastao and D. F. M. Torres: Fractional optimal control in the sense of Caputo and the fractional Noether's theorem. Int. Math. Forum 3 (2008), 10, 479-493.   CrossRef
  22. A. N. Kochubei: Fractional order diffusion. Diff. Equations 26 (1990), 485-492.   CrossRef
  23. A. N. Kochubei: General fractional calculus, evolution equations, and renewal processes. Integr. Equat. Oper. Theory 71 (2011), 583-600.   DOI:10.1007/s00020-011-1918-8
  24. W. Kotarski, H. A. El-Saify and G. M. Bahaa: 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
  25. J. L. Lions: Optimal Control Of Systems Governed By Partial Differential Equations. Springer-Verlag, Band 170 (1971).   CrossRef
  26. J. L. Lions and E. Magenes: Non-Homogeneous Boundary Value Problem and Applications. Springer-Verlag, New York 1972.   DOI:10.1007/978-3-642-65217-2
  27. Y. Liu, W. Rundell and M. Yamamoto: 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
  28. Z. Liu, Sh. Zeng and Y. Bai: 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
  29. Yu. Luchko: 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
  30. Yu. Luchko: Boundary value problems for the generalized time fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12 (2009), 409-422.   CrossRef
  31. Yu. Luchko: 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
  32. Yu. Luchko: 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
  33. Yu. Luchko and M. Yamamoto: 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
  34. I. Matychyna and V. Onyshchenkob: On time-optimal control of fractional-order systems. J. Comput. Appl. Math. 339 (2018), 245-257.   DOI:10.1016/j.cam.2017.10.016
  35. G. M. Mophou: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61 (2011), 68-78.   DOI:10.1016/j.camwa.2010.10.030
  36. G. M. Mophou: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62 (2011), 1413-1426.   DOI:10.1016/j.camwa.2011.04.044
  37. K. B. Oldham and J. Spanier: The Fractional Calculus. Academic Press, New York 1974.   CrossRef
  38. M. Protter and H. Weinberger: Maximum Principles in Differential Equations. Prentice Hall, Englewood Clifs, 1967.   DOI:10.1007/978-1-4612-5282-5
  39. H. Ye, F. Liu, V. Anh and I. Turner: 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