Variable-order Caputo fractional optimal control problems: optimality conditions and a forward–backward numerical method

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Abstract

This paper investigates an optimal control problem governed by variable-order Caputo fractional dynamics. The main difficulty arises from the fact that, for a type-I variable-order Caputo derivative, the corresponding adjoint operator is not obtained by a direct replacement of the constant fractional order in the standard right-sided Riemann–Liouville derivative. Instead, it must be derived from the dual kernel induced by the variable-order Caputo operator. Based on this observation, we formulate a right-sided variable-order adjoint operator and establish the associated integration-by-parts formula. Under suitable regularity assumptions, first-order necessary optimality conditions are derived for a class of quadratic optimal control problems with nonlinear variable-order fractional dynamics. The resulting optimality system consists of a forward variable-order Caputo state equation, a backward adjoint equation involving the dual-kernel operator, a stationarity condition, and a natural transversality condition. A predictor–corrector forward–backward sweep method is then developed for the numerical approximation of the coupled system. The adjoint equation is treated by a backward Volterra-type approximation motivated by the fractional memory structure. Numerical examples, including linear-quadratic, nonlinear scalar, and coupled nonlinear systems, illustrate the feasibility and mesh-refinement behavior of the proposed approach.

Keywords

variable-order fractional calculus / Caputo derivative / fractional optimal control / adjoint equation / integration by parts / forward–backward sweep method / predictor–corrector method

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References

[1] K. Diethelm, The Analysis of Fractional Differential Equations , Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2

[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations , Elsevier, Amsterdam, 2006.

[3] K. B. Oldham, J. Spanier, The Fractional Calculus , Academic Press, New York, 1974.

[4] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999.

[5] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications , Gordon and Breach, London, 1993.

[6] H. Q. Ma, C. H. Xie, W. R. Wang, L. Y. Zhou, Z. H. Liu, T. Song, et al., Numerical modeling of wet milling process by an improved CFD-DEM-VOF model, MetaResource, 2 (2025), 33-55. https://doi.org/10.23919/METAR.2025.000003

[7] Y. Tian, X. Y. Guo, Q. Zhang, Z. C. Shen, X. W. Dong, C. Z. Li, et al., A Lagrangian mesh-free multi-phase model for simulating complex dynamics of bubbly flows, MetaResource, 2 (2025), 11-32. https://doi.org/10.23919/METAR.2025.000002

[8] C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 515 (2003), 692-703. https://doi.org/10.1002/andp.200351511-1203

[9] C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57-98. https://doi.org/10.1023/A:1016586905654

[10] H. G. Sun, W. Chen, Y. Q. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), 4586-4592. https://doi.org/10.1016/j.physa.2009.07.024

[11] H. Hassani, Z. Avazzadeh, A. Turan-Dincel, P. Rahimkhani, A novel optimisation strategy for solving optimal control of variable-order fractional dynamic systems with nonlocal boundary conditions, Int. J. Syst. Sci., 57 (2026), 1703-1717. https://doi.org/10.1080/00207721.2025.2536210

[12] M. Alipour, S. Soradi-Zeid, Optimal control governed by nonlinear variable-order fractional integro-differential equations through generalized Dickson polynomial expansions, J. Comput. Appl. Math., 473 (2026), 116917. https://doi.org/10.1016/j.cam.2025.116917

[13] X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522-1545. https://doi.org/10.1093/imanum/draa013

[14] M. H. Heydari, Z. Avazzadeh, A hybrid method for variable-order fractional 2D optimal control problems on an unbounded domain, Eng. Comput., 38 (2022), 3237-3249. https://doi.org/10.1007/s00366-021-01287-w

[15] M. M. A. Hasan, Optimal control problem of variable-order delay system of advertising procedure: numerical treatment, Discrete Contin. Dyn. Syst. - Ser. S, 15 (2022), 1247-1268. https://doi.org/10.3934/dcdss.2021085

[16] N. Kumar, M. Mehra, Legendre wavelet method for solving variable-order nonlinear fractional optimal control problems with variable-order fractional Bolza cost, Asian J. Control, 25 (2023), 2122-2138. https://doi.org/10.1002/asjc.2856

[17] F. Soufivand, F. Soltanian, K. Mamehrashi, A numerical approach for solving a class of two-dimensional variable-order fractional optimal control problems using Gegenbauer operational matrix, IMA J. Math. Control Inf., 40 (2023), 1-19. https://doi.org/10.1093/imamci/dnac031

[18] A. Singh, A. Kanaujiya, J. Mohapatra, Chelyshkov wavelet method for solving multidimensional variable order fractional optimal control problem, J. Appl. Math. Comput., 70 (2024), 3135-3160. https://doi.org/10.1007/s12190-024-02083-7

[19] N. H. Sweilam, F. Megahed, S. A. Shatta, D. Baleanu, Optimal control for a variable-order diffusion-wave equation with a reaction term: a numerical study, Partial Differ. Equations Appl. Math., 10 (2024), 100658. https://doi.org/10.1016/j.padiff.2024.100658

[20] X. Zheng, Z. Yang, W. Li, H. Wang, A time-fractional mean field control modeling subdiffusive advective transport, SIAM J. Sci. Comput., 45 (2023), B884-B905. https://doi.org/10.1137/22M1527726

[21] M. Athans, P. L. Falb, Optimal Control: An Introduction to the Theory and Its Applications , Dover Publications, New York, 2007.

[22] A. E. Bryson, Applied Optimal Control , Routledge, Cambridge, 1975.

[23] D. E. Kirk, Optimal Control Theory: An Introduction , Dover Publications, New York, 2004.

[24] D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction , Princeton University Press, Princeton, 2012. https://doi.org/10.1515/9781400842643

[25] O. P. Agrawal, General formulation for the numerical solution of optimal control problems, Int. J. Control, 50 (1989), 627-638. https://doi.org/10.1080/00207178908953385

[26] O. P. Agrawal, O. Defterli, D. Baleanu, Fractional optimal control problems with several state and control variables, J. Vib. Control, 16 (2010), 1967-1976. https://doi.org/10.1177/1077546309353361

[27] T. Akbarian, M. Keyanpour, A new approach to the numerical solution of fractional order optimal control problems, Appl. Appl. Math., 8 (2013), 523-534. Available from: https://digitalcommons.pvamu.edu/aam/vol8/iss2/12.

[28] A. Alizadeh, S. Effati, An iterative approach for solving fractional optimal control problems, J. Vib. Control, 24 (2018), 18-36. https://doi.org/10.1177/1077546316633391

[29] A. Jajarmi, D. Baleanu, On the fractional optimal control problems with a general derivative operator, Asian J. Control, 23 (2021), 1062-1071. https://doi.org/10.1002/asjc.2282

[30] P. Sahu, R. S. Saha, Comparison on wavelets techniques for solving fractional optimal control problems, J. Vib. Control, 24 (2018), 1185-1201. https://doi.org/10.1177/1077546316659611

[31] N. Singha, C. Nahak, An efficient approximation technique for solving a class of fractional optimal control problems, J. Optim. Theory Appl., 174 (2017), 785-802. https://doi.org/10.1007/s10957-017-1143-y