Necessary optimality conditions for quasi-singular controls for systems with Caputo fractional derivatives

• Autorzy: Yusubov Shakir Sh. , Mahmudov Elimhan N.

Identyfikatory

DOI

10.24425/acs.2023.146955

• Języki publikacji: EN

Abstrakty

EN

In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. First we get the linearized maximum principle. Further, the concept of a quasi-singular control is introduced and, on this basis, an analogue of the Legendre-Clebsch conditions is obtained. When the analogue of Legendre-Clebsch condition degenerates, a necessary high-order optimality condition is derived. An illustrative example is considered.

Słowa kluczowe

EN

fractional derivative; fractional optimal control; necessary optimality condition

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2023
• Tom: Vol. 33, No. 3
• Strony: 463--496
• Opis fizyczny: Bibliogr. 49 poz., wzory

Twórcy

autor

Yusubov Shakir Sh.

• Department of Mechanics and Mathematics, Baku State University, Baku, Azerbaijan

autor

Mahmudov Elimhan N.

• Department of Mathematics, Istanbul Technical University, Istanbul, Turkey
• Azerbaijan National Aviation Academy, Baku, Azerbaijan

• https://orcid.org/0000-0003-2879-6154

Bibliografia

• [1] O.P. Agrawal: A general formulation and solution scheme for fractional optimal control problems. Nonlin Dynamics, 38 (2004), 323-337. DOI: 10.1007/s11071-004-3764-6.
• [2] N.U. Ahmed and C.D. Charalambous: Filtering for linear systems driven by fractional brownian motion. SIAM Journal on Control and Optimization, 41(1), (2002), 313-330. DOI: 10.1137/S0363012900368715.
• [3] R. Almeida, R. Kamocki, A.B. Malinowska and T. Odzijewicz: On the necessary optimality conditions for the fractional Cucker-Smale optimal control problem. Communications in Nonlinear Science and Numerical Simulation, 96 (2021), 1-22. DOI: 10.1016/j.cnsns.2020.105678.
• [4] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo: Fractional Calculus Models and Numerical Methods. World Scientific Publishing Co. Pte. Ltd. Hackensack, New York. 2012.
• [5] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications, 338(2), (2008), 1340-1350. DOI: 10.1016/j.jmaa.2007.06.021.
• [6] M. Bergounioux and L. Bourdin: Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints. ESAIM: Control, Optimization and Calculus of Variations, 26, (2020), 1-38. DOI: 10.1051/cocv/2019021.
• [7] P. Bogdan and R. Marculescu: A fractional calculus approach to modeling fractal dynamic games. 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA, (2011), 255-260. DOI: 10.1109/CDC.2011.6161323.
• [8] V. Capasso, S. Anita and V. Arnăutu: An Introduction to Optimal Control Problems in Life Sciences and Economics. Birkhauser, Boston. 2010.
• [9] A. Carpinteri and F. Mainardi: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin. 1997.
• [10] T. Chiranjeevi, R.K. Biswas, R. Devarapalli, N.R. Babu and F.P. Garcia Márquez: On optimal control problem subject to fractional order discrete time singular systems. Archives of Control Sciences, 31(4), (2021), 849-863. DOI: 10.24425/acs.2021.139733.
• [11] K. Diethelm: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition using Differential Operators of Caputo Type. Lecture Notes in Matematics, Spinger-Verlag, Berlin. 2010.
• [12] T.E. Duncan and B. Pasik-Duncan: Linear-quadratic fractional gaussian control. SIAM Journal of Control, 51(6), (2013), 4504-4519. DOI: 10.1137/120877283.
• [13] T. Fajraoui, B. Ghanmi, F. Mabrouk and F. Omri: Mittag-Leffler stability analysis of a class of homogeneous fractional systems. Archives of Control Sciences, 31(2), (2021), 401-415. DOI: 10.24425/acs.2021.137424.
• [14] O.S. Fard, J. Soolaki and D.F.M. Torres: A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - Series S, 11(1), (2018), 59-76. DOI: 10.3934/dcdss.2018004.
• [15] R. Gabasov and F.M. Kirillova: High-order necessary conditions for optimality. SIAM Journal of Control, 10 (1972), 127-168. DOI: 10.1137/0310012.
• [16] M.I. Gomoyunov: Optimal control problems with a fixed terminal time in linear fractional-order systems. Archives of Control Sciences, 30(4), (2020), 721-744. DOI: 10.24425/acs.2020.135849.
• [17] W.M. Haddad and A. L’Afflitto: Finite-time partial stability theory and fractional Lyapunov differential inequalities. 2015 American Control Conference (ACC), Chicago, IL, USA, (2025), 5347-5352. DOI: 10.1109/ ACC.2015.7172175.
• [18] D. Henry: Geometric Theory of Similinear Parabolic Prtial Differential Equations. Springer-Verlag, Berlin, Gremany, New York, NY, USA. 1989.
• [19] D. Idczak and S. Walczak: On a linear-quadratic problem with Caputo derivative. Opuscula Mathematica, 36 (2016), 49-68. DOI: 10.7494/OpMath.2016.36.1.49.
• [20] T. Kaczorek: Stability of fractional positive nonlinear systems. Archives of Control Sciences, 25(4), (2015), 491-496. DOI: 10.1515/acsc-2015-0031.
• [21] R. Kamocki: Pontryagin maximum principle for fractional ordinary optimal control problems. Mathematical Methods in the Applied Sciences, 37(11), (2014), 1668-1686. DOI: 10.1002/mma.2928.
• [22] A.A. Kilbas, H.M. Srivastava and J.J.Trujillo: Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Mathatics Studies, Elsevier Science B.V. Amsterdam. 2006.
• [23] J. Klamka: Local controllability of fractional discrete-time semilinear systems. Acta Mechanica et Automatica, 5(2), (2012), 55-58.
• [24] A.J. Krener: The high order maximal principle and its application to singular extremals. SIAM Journal of Control, 15(2), (1977), 256-293. DOI: 10.1137/0315019.
• [25] P. Kulczycki, J. Korbicz and J. Kacprzyk: Fractional Dynamical Systems: Methods, Algorithms and Applications. Springer, 2022.
• [26] S. Ladaci, J.J. Loiseau and A. Charef: Adaptive internal model control with fractional order parameter. International Journal of Adaptive Control and Signal Processing, 24(11), (2010), 944-960. DOI: 10.1002/acs.1175.
• [27] I. Lasiecka and R. Triggiani: Domains of Fractional Powers of Matrix-valued Operators: A General Approach. Operator Theory: Adv. Appl. Springer Int. Publish. 250, 2015, 297-309.
• [28] Ch. Liu, W. Sun and X. Yi: Optimal control of a fractional smoking system. Journal of Industrial and Management Optimization, 19(4), (2023), 29362954. DOI: 10.3934/jimo.2022071.
• [29] W. Li, S.S. Wang and V. Rehbock: Numerical solution of fractional optimal control. Journal of Optimization Theory and Applications, 180 (2019), 556-573. DOI: 10.1007/s10957-018-1418-y.
• [30] M.S. Mahmoud and Yuanqing Xia: Applied Control Systems Design. Springer, 2012.
• [31] E.N. Mahmudov: Optimal control of Cauchy problem for first-order discrete and partial differential inclusions. Journal of Dynamical and Control Systems, 15 (2009), 587-610. DOI: 10.1007/s10883-009-9073-0.
• [32] E.N. Mahmudov: Optimal control of higher order differential inclusions with functional constraints. ESAIM: Control, Optimisation and Calculus of Variations, 26 (2020), 1-23. DOI: 10.1051/cocv/2019018.
• [33] E.N. Mahmudov: Approximation and Optimization of Discrete and Differential Inclusions. Boston MA, USA, Elsevier. 2011.
• [34] E.N. Mahmudov and Sh.Sh. Yusubov: Nonlocal boundary value problems for hyperbolic equations with a Caputo fractional derivative. Journal of Computational and Applied Mathematics, 398 (2021), 1-15. DOI: 10.1016/j.cam.2021.113709.
• [35] E.N. Mahmudov and M.J. Mardanov: Optimization of the Nicoletti boundary value problem for second-order differential inclusions. Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 46 (2023), 3-15. DOI: 10.30546/2409-4994.2023.49.1.3.
• [36] I. Matychyn and V. Onyshchenko: Optimal control of linear systems with fractional derivatives. Fractional Calculus and Applied Analysis, 21 (2018), 134-150. DOI: 10.1515/fca-2018-0009.
• [37] K.S. Miller and B. Ross: An İntroduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Intersience Publication. John Wiley And Sons. Inc. New York. 1993.
• [38] W. Mitkowski and A. Obraczka: Simple identification of fractional differential equation. Solid State Phenomena, 180 (2012), 331-338. DOI: 10.4028/www.scientific.net/SSP.180.331.
• [39] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishenko: Mathematical Theory of Optimal Processes. Nauka, Moscow, Gordon and Breach, New York. 1986.
• [40] A. Ruszewski: Stability of discrete-time fractional linear systems with delays. Archives of Control Sciences, 29(3), (2019), 549-567. DOI: 10.24425/acs.2019.130205.
• [41] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional integrals and derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon, Switzerland. 1993.
• [42] H.T. Tuan, A. Czornik, J.J. Nieto and M. Niezabitowski: Global attractivity for some classes of Riemann-Liouville fractional differential systems. Journal of Integral Equations and Applications, 31(2), (2017). DOI: 10.1216/JIE-2019-31-2-265.
• [43] L. Yulianti, A. Nazra, Zulakmal, A. Bahar and Muhafzan: On discounted LQR control problem for disturbanced singular system. Archives of Control Sciences, 29(1), (2019), 147-156. DOI: 10.24425/acs.2019.127528.
• [44] Sh.Sh. Yusubov: Necessary conditions of optimality for systems with impulsive actions. Computational Mathematics and Mathematical Physics, 45 (2005), 222-226. eLIBRARY ID: 27848950
• [45] Sh.Sh. Yusubov: Necessary optimality Cconditions for singular controls. Computational Mathematics and Mathematical Physics, 47 (2007), 1446-1451. DOI: 10.1134/S0965542507090060.
• [46] Sh.Sh. Yusubov: On an optimality of the singular with respect to components controls in the Goursat-Dourboux systems. Problemy Upravleniya, 5, (2014), 2-6.
• [47] Sh.Sh. Yusubov: Boundary value problems for hyperbolic equations with a Caputo fractional derivative. Advanced Mathematical Models & Applications, 5(2), (2020), 192-204.
• [48] Sh.Sh. Yusubov and E.N. Mahmudov: Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial and Management Optimization, 19(1), (2023), 246-264. DOI: 10.3934/jimo.2021182.
• [49] Sh.Sh. Yusubov and E.N. Mahmudov: Necessary and sufficient optimality conditions for fractional Fornasini-Marchesini model. Journal of Industrial and Management Optimization, 19(10), (2023) 7221-7244. DOI: 10.3934/jimo.2022260.