Yıldız, Tuğba Akman; Jajarmi, Amin; Yıldız, Burak; Baleanu, Dumitru New aspects of time fractional optimal control problems within operators with nonsingular kernel. (English) Zbl 1439.49010 Discrete Contin. Dyn. Syst., Ser. S 13, No. 3, 407-428 (2020). Summary: This paper deals with a new formulation of time fractional optimal control problems governed by Caputo-Fabrizio (CF) fractional derivative. The optimality system for this problem is derived, which contains the forward and backward fractional differential equations in the sense of CF. These equations are then expressed in terms of Volterra integrals and also solved by a new numerical scheme based on approximating the Volterra integrals. The linear rate of convergence for this method is also justified theoretically. We present three illustrative examples to show the performance of this method. These examples also test the contribution of using CF derivative for dynamical constraints and we observe the efficiency of this new approach compared to the classical version of fractional operators. Cited in 38 Documents MSC: 49J21 Existence theories for optimal control problems involving relations other than differential equations 34A08 Fractional ordinary differential equations 34H05 Control problems involving ordinary differential equations 49M25 Discrete approximations in optimal control 49K21 Optimality conditions for problems involving relations other than differential equations Keywords:optimal control; nonsingular kernel; fractional calculus; error estimates; Volterra integrals PDFBibTeX XMLCite \textit{T. A. Yıldız} et al., Discrete Contin. Dyn. Syst., Ser. S 13, No. 3, 407--428 (2020; Zbl 1439.49010) Full Text: DOI References: [1] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Advances in Difference Equations, 2017 (2017), Paper No. 313, 11 pp. · Zbl 1444.26003 [2] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, Journal of Inequalities and Applications, 2017 (2017), Paper No. 130, 11 pp. · Zbl 1368.26003 [3] T. Abdeljawad; Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality, Journal of Computational and Applied Mathematics, 339, 218-230 (2018) · Zbl 1472.39006 [4] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Advances in Difference Equations, 2016 (2016), Paper No. 232, 18 pp. · Zbl 1419.34211 [5] T. Abdeljawad; D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, Journal of Nonlinear Sciences and Applications, 10, 1098-1107 (2017) · Zbl 1412.47086 [6] T. Abdeljawad; D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos, Solitons & Fractals, 102, 106-110 (2017) · Zbl 1374.26011 [7] T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Advances in Difference Equations, 2017 (2017), Paper No. 78, 9 pp. · Zbl 1422.39048 [8] T. Abdeljawad; D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Reports on Mathematical Physics, 80, 11-27 (2017) · Zbl 1384.26025 [9] T. Abdeljawad; F. Madjidi, Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order \(2 < α < 5/2\), The European Physical Journal Special Topics, 226, 3355-3368 (2017) [10] O. Agrawal, General formulation for the numerical solution of optimal control problems, International Journal of Control, 50, 627-638 (1989) · Zbl 0679.49031 [11] M. Al-Refai and T. Abdeljawad, Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel, Advances in Difference Equations, 2017 (2017), Paper No. 315, 12 pp. · Zbl 1444.35143 [12] B. S. Alkahtani; O. J. Algahtani; R. S. Dubey; P. Goswam, The solution of modified fractional Bergman’s minimal blood glucose-insulin model, Entropy, 19, 114 (2017) [13] A. Atangana; D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20, 763-769 (2016) [14] R. L. Bagley; P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27, 201-210 (1983) · Zbl 0515.76012 [15] D. Baleanu; A. Jajarmi; M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel, Journal of Optimization Theory and Applications, 175, 718-737 (2017) · Zbl 1383.49030 [16] R. K. Biswas; S. Sen, Fractional optimal control problems with specified final time, Journal of Computational and Nonlinear Dynamics, 6, 021009 (2011) [17] M. Caputo; M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1, 1-13 (2015) [18] M. Caputo; M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2, 1-11 (2016) [19] S. Choi; E. Jung; S.-M. Lee, Optimal intervention strategy for prevention tuberculosis using a smoking-tuberculosis model, Journal of Theoretical Biology, 380, 256-270 (2015) · Zbl 1343.92466 [20] G. M. Coclite; M. Garavello; L. V. Spinolo, Optimal strategies for a time-dependent harvesting problem, Discrete & Continuous Dynamical Systems-S, 11, 865-900 (2018) · Zbl 1405.35099 [21] E. F. Doungmo Goufo and S. Mugisha, On analysis of fractional Navier-Stokes equations via nonsingular solutions and approximation, Mathematical Problems in Engineering, 2015 (2015), Art. ID 212760, 8 pp. · Zbl 1394.35551 [22] N. Ejlali; S. M. Hosseini, A pseudospectral method for fractional optimal control problems, Journal of Optimization Theory and Applications, 174, 83-107 (2017) · Zbl 1377.49019 [23] M. Enelund; P. Olsson, Damping described by fading memory-analysis and application to fractional derivative models, International Journal of Solids and Structures, 36, 939-970 (1999) · Zbl 0936.74023 [24] J. Fujioka, A. Espinosa, R. F. Rodríguez and B. A. Malomed, Radiating subdispersive fractional optical solitons, Chaos, 24 (2014), 033121, 11pp. · Zbl 1374.35375 [25] D.-p. Gao; N.-j. Huang, Optimal control analysis of a tuberculosis model, Applied Mathematical Modelling, 58, 47-64 (2018) · Zbl 1464.92143 [26] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000. · Zbl 0998.26002 [27] J. Hristov, Derivation of fractional Dodson’s equation and beyond: Transient mass diffusion with a non-singular memory and exponentially fading-out diffusivity, Progress in Fractional Differentiation and Applications, 3, 255-270 (2017) [28] J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20, 757-762 (2016) [29] J. Hristov, Derivatives with non-singular kernels from the caputo-fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus. Sharjah: Bentham Science Publishers, 1, 269-341 (2018) [30] C. Ionescu; K. Desager; R. De Keyser, Fractional order model parameters for the respiratory input impedance in healthy and in asthmatic children, Computer Methods and Programs in Biomedicine, 101, 315-323 (2011) [31] C. M. Ionescu; R. De Keyser, Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease, IEEE Transactions on Biomedical Engineering, 56, 978-987 (2009) [32] S. Jahanshahi; D. F. Torres, A simple accurate method for solving fractional variational and optimal control problems, Journal of Optimization Theory and Applications, 174, 156-175 (2017) · Zbl 1377.49016 [33] F. Jarad; T. Abdeljawad; D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218, 9234-9240 (2012) · Zbl 1244.49028 [34] Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38 (2009), 571-581, URL http://dx.doi.org/10.1007/s00158-008-0307-7. · Zbl 1274.49035 [35] T. Kaczorek, Reachability of fractional continuous-time linear systems using the Caputo-Fabrizio derivative, in ECMS, 2016, 53-58. [36] T. Kaczorek; K. Borawski, Fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative, International Journal of Applied Mathematics and Computer Science, 26, 533-541 (2016) · Zbl 1347.93147 [37] C. K. Kwuimy; G. Litak; C. Nataraj, Nonlinear analysis of energy harvesting systems with fractional order physical properties, Nonlinear Dynamics, 80, 491-501 (2015) [38] C. Li; F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34, 149-179 (2013) · Zbl 1267.65094 [39] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, vol. 24, CRC Press, 2015. · Zbl 1326.65033 [40] J. Liouville, M \(é\) moire: Sur quelques questions de g \(é\) om \(é\) trie et de m \(é\) canique, et sur un nouveau genre de calcul pour r \(é\) soudre ces questions, J l’Ecole Polytéch, 13, 1-66 (1832) [41] A. Lotfi; M. Dehghan; S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Computers & Mathematics with Applications, 62, 1055-1067 (2011) · Zbl 1228.65109 [42] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Redding, 2006. [43] J. P. Mateus; P. Rebelo; S. Rosa; C. M. Silva; D. F. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete & Continuous Dynamical Systems-S, 11, 1179-1199 (2018) · Zbl 1408.49046 [44] V. Morales-Delgado; J. Gómez-Aguilar; M. Taneco-Hernandez, Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives, The European Physical Journal Plus, 132, 527 (2017) [45] A. Nemati; S. A. Yousefi, A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix, IMA Journal of Mathematical Control and Information, 34, 1079-1097 (2017) · Zbl 1400.49033 [46] T. Ohtsuka; K. Shirakawa; N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy, Discrete Contin. Dyn. Syst. Ser. S, 5, 159-181 (2012) · Zbl 1239.49004 [47] I. Petráš; R. L. Magin, Simulation of drug uptake in a two compartmental fractional model for a biological system, Communications in Nonlinear Science and Numerical Simulation, 16, 4588-4595 (2011) · Zbl 1229.92043 [48] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. · Zbl 0924.34008 [49] S. Pooseh; R. Almeida; D. F. Torres, Fractional order optimal control problems with free terminal time, Journal of Industrial & Management Optimization, 10, 363-381 (2014) · Zbl 1278.26013 [50] J. K. Popović; M. T. Atanacković; A. S. Pilipović; M. R. Rapaić; S. Pilipović; T. M. Atanacković, A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac, Journal of Pharmacokinetics and Pharmacodynamics, 37, 119-134 (2010) [51] S. G. Samko, A. A. Kilbas, O. I. Marichev et al., Fractional Integrals and Derivatives, vol. 1993, 1993. · Zbl 0818.26003 [52] N. A. Sheikh; F. Ali; M. Saqib; I. Khan; S. A. A. Jan; A. S. Alshomrani; M. S. Alghamdi, Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7, 789-800 (2017) [53] A. A. Tateishi; H. V. Ribeiro; E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physicss, 5, 52 (2017) [54] D. Verotta, Fractional dynamics pharmacokinetics-pharmacodynamic models, Journal of Pharmacokinetics and Pharmacodynamics, 37, 257-276 (2010) [55] J. Wang; Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Analysis: Real World Applications, 12, 262-272 (2011) · Zbl 1214.34010 [56] S. H. Weinberg, Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model, PloS One, 10, e0126629 (2015) [57] D. Xue; L. Bai, Numerical algorithms for Caputo fractional-order differential equations, International Journal of Control, 90, 1201-1211 (2017) · Zbl 1369.65085 [58] A.-M. Yang; Y. Han; J. Li; W.-X. Liu, On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Thermal Science, 20, 717-721 (2016) [59] X.-J. Yang; F. Gao; J. Machado; D. Baleanu, A new fractional derivative involving the normalized Sinc function without singular kernel, The European Physical Journal Special Topics, 226, 3567-3575 (2017) [60] J. Zhang, X. Ma and L. Li, Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives, Advances in Difference Equations, 2017 (2017), Paper No. 357, 14 pp. · Zbl 1444.26006 [61] Y. Zhang; H. Sun; H. H. Stowell; M. Zayernouri; S. E. Hansen, A review of applications of fractional calculus in Earth system dynamics, Chaos, Solitons & Fractals, 102, 29-46 (2017) · Zbl 1374.86028 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.