A Pontryagin maximum principle for optimal control problems involving generalized distributional-order derivatives

Fátima Cruz; Ricardo Almeida; Natália Martins; Fátima Cruz; Ricardo Almeida; Natália Martins

doi:10.3934/math.2025539

AIMS Mathematics

2025, Volume 10, Issue 5: 11939-11956. doi: 10.3934/math.2025539

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Research article

A Pontryagin maximum principle for optimal control problems involving generalized distributional-order derivatives

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

Received: 03 April 2025 Revised: 08 May 2025 Accepted: 14 May 2025 Published: 22 May 2025
MSC : 26A33, 49K05

In this work, we extended fractional optimal control (OC) theory by proving a version of Pontryagin's maximum principle and establishing sufficient optimality conditions for an OC problem. The dynamical system constraint in the OC problem under investigation is described by a generalized fractional derivative: the left-sided Caputo distributed-order fractional derivative with an arbitrary kernel. This approach provides a more versatile representation of dynamic processes, accommodating a broader range of memory effects and hereditary properties inherent in diverse physical, biological, and engineering systems.
- fractional calculus,
- fractional optimal control,
- Pontryagin maximum principle
Citation: Fátima Cruz, Ricardo Almeida, Natália Martins. A Pontryagin maximum principle for optimal control problems involving generalized distributional-order derivatives[J]. AIMS Mathematics, 2025, 10(5): 11939-11956. doi: 10.3934/math.2025539

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Abstract

In this work, we extended fractional optimal control (OC) theory by proving a version of Pontryagin's maximum principle and establishing sufficient optimality conditions for an OC problem. The dynamical system constraint in the OC problem under investigation is described by a generalized fractional derivative: the left-sided Caputo distributed-order fractional derivative with an arbitrary kernel. This approach provides a more versatile representation of dynamic processes, accommodating a broader range of memory effects and hereditary properties inherent in diverse physical, biological, and engineering systems.

References

[1]	O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38 (2004), 323–337. https://doi.org/10.1007/s11071-004-3764-6 doi: 10.1007/s11071-004-3764-6
[2]	O. P. Agrawal, A formulation and a numerical scheme for fractional optimal control problems, IFAC Proc. Vol., 39 (2006), 68–72. https://doi.org/10.3182/20060719-3-PT-4902.00011 doi: 10.3182/20060719-3-PT-4902.00011
[3]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[4]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[5]	M. Bergounioux, L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), 35. https://doi.org/10.1051/cocv/2019021 doi: 10.1051/cocv/2019021
[6]	A. Boukhouima, K. Hattaf, E. M. Lotfi, M. Mahrouf, D. F. M. Torres, N. Yousfi, Lyapunov functions for fractional-order systems in biology: Methods and applications, Chaos Solitons Fractals, 140 (2020), 110224. https://doi.org/10.1016/j.chaos.2020.110224 doi: 10.1016/j.chaos.2020.110224
[7]	B. van Brunt, The calculus of variations, New York: Springer, 2003. https://doi.org/10.1007/b97436
[8]	M. Caputo, Mean fractional-order-derivatives differential equations and filters, Ann. Univ. Ferrara, 41 (1995), 73–84. https://doi.org/10.1007/BF02826009 doi: 10.1007/BF02826009
[9]	J. F. Chen, H. T. Quan, Optimal control theory for maximum power of Brownian heat engines, Phys. Rev. E, 110 (2024), L042105. https://doi.org/10.1103/PhysRevE.110.L042105 doi: 10.1103/PhysRevE.110.L042105
[10]	F. Cruz, R. Almeida, N. Martins, Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels, AIMS Mathematics, 6 (2021), 5351–5369. https://doi.org/10.3934/math.2021315 doi: 10.3934/math.2021315
[11]	A. N. Daryina, A. I. Diveev, D. Yu. Karamzin, F. L. Pereira, E. A. Sofronova, R. A. Chertovskikh, Regular approximations of the fastest motion of mobile robot under bounded state variables, Comput. Math. Math. Phys., 62 (2022), 1539–1558. https://doi.org/10.1134/S0965542522090093 doi: 10.1134/S0965542522090093
[12]	K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
[13]	W. Ding, S. Patnaik, S. Sidhardh, F. Semperlotti, Applications of distributed-order fractional operators: A review, Entropy, 23 (2021), 110. https://doi.org/10.3390/e23010110 doi: 10.3390/e23010110
[14]	G. S. F. Frederico, D. F. M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether's theorem, Int. Math. Forum, 3 (2008), 479–493.
[15]	G. S. F. Frederico, D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dyn., 53 (2008), 215–222. https://doi.org/10.1007/s11071-007-9309-z doi: 10.1007/s11071-007-9309-z
[16]	J. J. Gasimov, J. A. Asadzade, N. I. Mahmudov, Pontryagin maximum principle for fractional delay differential equations and controlled weakly singular Volterra delay integral equations, Qual. Theory Dyn. Syst., 23 (2024), 213. https://doi.org/10.1007/s12346-024-01049-1 doi: 10.1007/s12346-024-01049-1
[17]	H. P. Geering, Optimal control with engineering applications, Heidelberg: Springer Berlin, 2007. https://doi.org/10.1007/978-3-540-69438-0
[18]	J. Jiang, The distributed order models to characterize the flow and heat transfer of viscoelastic fluid between coaxial cylinders, Phys. Scr., 99 (2024), 015233. https://doi.org/10.1088/1402-4896/ad1379 doi: 10.1088/1402-4896/ad1379
[19]	R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668–1686. https://doi.org/10.1002/mma.2928 doi: 10.1002/mma.2928
[20]	R. Kamocki, Pontryagin's maximum principle for a fractional integro-differential Lagrange problem, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107598. https://doi.org/10.1016/j.cnsns.2023.107598 doi: 10.1016/j.cnsns.2023.107598
[21]	S. L. Khalaf, K. K. Kassid, A. R. Khudair, A numerical method for solving quadratic fractional optimal control problems, Results Control Optim., 13 (2023), 100330. https://doi.org/10.1016/j.rico.2023.100330 doi: 10.1016/j.rico.2023.100330
[22]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, In: North-Holland mathematics studies, Elsevier, 204 (2006), 1–523.
[23]	D. Kumar, J. Singh, Fractional calculus in medical and health science, Boca Raton: CRC Press, 2020. https://doi.org/10.1201/9780429340567
[24]	S. Lenhart, J. T. Workman, Optimal control applied to biological models, New York: Chapman & Hall/CRC, 2007. https://doi.org/10.1201/9781420011418
[25]	W. Li, S. Wang, V. Rehbock, Numerical solution of fractional optimal control, J. Optim. Theory Appl., 180 (2019), 556–573. https://doi.org/10.1007/s10957-018-1418-y doi: 10.1007/s10957-018-1418-y
[26]	D. Liberzon, Calculus of variations and optimal control theory: A concise introduction, Princeton University Press, 2012.
[27]	A. Lotfi, M. Dehghan, S. A. Yousefi, A numerical technique for solving fractional optimal control problems, Comput. Math. Appl., 62 (2011), 1055–1067. https://doi.org/10.1016/j.camwa.2011.03.044 doi: 10.1016/j.camwa.2011.03.044
[28]	F. Ndairou, I. Area, J. J. Nieto, C. J. Silva, D. F. M. Torres, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos Solitons Fract., 144 (2021), 110652. https://doi.org/10.1016/j.chaos.2021.110652 doi: 10.1016/j.chaos.2021.110652
[29]	F. Ndairou, D. F. M. Torres, Distributed-order non-local optimal control, Axioms, 9 (2020), 124. https://doi.org/10.3390/axioms9040124 doi: 10.3390/axioms9040124
[30]	F. Ndairou, D. F. M. Torres, Pontryagin maximum principle for distributed-order fractional systems, Mathematics, 9 (2021), 1883. https://doi.org/10.3390/math9161883 doi: 10.3390/math9161883
[31]	K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41 (2010), 9–12. https://doi.org/10.1016/j.advengsoft.2008.12.012 doi: 10.1016/j.advengsoft.2008.12.012
[32]	S. F. Omid, S. Javad, D. F. M. Torres, A necessary condition of Pontryagin type for fuzzy fractional optimal control problems, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 59–76. https://doi.org/10.3934/dcdss.2018004 doi: 10.3934/dcdss.2018004
[33]	R. Ozarslan, E. Bas, D. Baleanu, B. Acay, Fractional physical problems including wind-influenced projectile motion with Mittag-Leffler kernel, AIMS Mathematics, 5 (2020), 467–481. https://doi.org/10.3934/math.2020031 doi: 10.3934/math.2020031
[34]	L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, New York: John Wiley & Sons, 1962.
[35]	F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581–3592. https://doi.org/10.1103/PhysRevE.55.3581 doi: 10.1103/PhysRevE.55.3581
[36]	R. T. Rockafellar, Convex analysis, Princeton University Press, 1997.
[37]	S. Rogosin, M. Karpiyenya, Fractional models for analysis of economic risks, Fract. Calc. Appl. Anal., 26 (2023), 2602–2617. https://doi.org/10.1007/s13540-023-00202-y doi: 10.1007/s13540-023-00202-y
[38]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach Science Publishers, 1993.
[39]	A. Seierstad, K. Sydsaeter, Optimal control theory with economic applications, 1 Eds., North Holland, 1987.
[40]	J. V. C. Sousa, E. C. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator, Differ. Equ. Appl., 11 (2019), 87–106. http://dx.doi.org/10.7153/dea-2019-11-02 doi: 10.7153/dea-2019-11-02
[41]	D. Tonon, M. S. Aronna, D. Kalise, Optimal control: Novel directions and applications, Springer International Publishing, 2017.
[42]	X. Yi, Z. Gong, C. Liu, H. T. Cheong, K. L. Teo, S. Wang, A control parameterization method for solving combined fractional optimal parameter selection and optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 141 (2025), 108462. https://doi.org/10.1016/j.cnsns.2024.108462 doi: 10.1016/j.cnsns.2024.108462

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