Time-optimal control of linear fractional systems with variable coefficients

• Autorzy: Matychyn Ivan , Onyshchenko Viktoriia

Identyfikatory

DOI

10.34768/amcs-2021-0025

• Języki publikacji: EN

Abstrakty

EN

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle are obtained. Theoretical results are supported by examples.

Słowa kluczowe

EN

fractional calculus; Riemann-Liouville derivative; variable coefficients; optimal control

PL

rachunek ułamkowy; pochodna ułamkowa Riemanna-Liouville'a; współczynnik zmienny; sterowanie optymalne

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2021
• Tom: Vol. 31, no. 3
• Strony: 375--386
• Opis fizyczny: Bibliogr. 35 poz., wykr.

Twórcy

autor

Matychyn Ivan

• Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Słoneczna 54, 10-710 Olsztyn, Poland

autor

Onyshchenko Viktoriia

• Institute of Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland

Bibliografia

• [1] Aumann, R.J. (1965). Integrals of set-valued functions, Journal of Mathematical Analysis and Applications 12(1): 1–12.
• [2] Baake, M. and Schlägel, U. (2011). The Peano–Baker series, Proceedings of the Steklov Institute of Mathematics 275(1): 155–159.
• [3] Balaska, H., Ladaci, S., Djouambi, A., Schulte, H. and Bourouba, B. (2020). Fractional order tube model reference adaptive control for a class of fractional order linear systems, International Journal of Applied Mathematics and Computer Science 30(3): 501–515, DOI: 10.34768/amcs-2020-0037.
• [4] Bergounioux, M. and Bourdin, L. (2020). Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM: Control, Optimisation and Calculus of Variations 26: 35, DOI: 10.1051/cocv/2019021.
• [5] Blagodatskikh, V.I. and Filippov, A.F. (1985). Differential inclusions and optimal control, Trudy Matematicheskogo Instituta Imeni VA Steklova 169: 194–252.
• [6] Bourdin, L. (2018). Cauchy–Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems, Differential and Integral Equations 31(7/8): 559–594.
• [7] Chikrii, A. and Eidelman, S. (2000). Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order, Cybernetics and System Analysis 36(3): 315–338.
• [8] Chikrii, A. and Matichin, I. (2008). Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann–Liouville, Caputo, and Miller–Ross, Journal of Automation and Information Sciences 40(6): 1–11.
• [9] Datsko, B. and Gafiychuk, V. (2018). Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point, Fractional Calculus and Applied Analysis 21(1): 237–253.
• [10] Datsko, B., Podlubny, I. and Povstenko, Y. (2019). Time-fractional diffusion-wave equation with mass absorption in a sphere under harmonic impact, Mathematics 7(5): 433.
• [11] Diethelm, K. (2010). The Analysis of Fractional Differential Equations, Springer, Berlin/Heidelberg.
• [12] Dzieliński, A. and Czyronis, P. (2013). Fixed final time and free final state optimal control problem for fractional dynamic systems—Linear quadratic discrete-time case, Bulletin of the Polish Academy of Sciences: Technical Sciences 61(3): 681–690.
• [13] Eckert, M., Nagatou, K., Rey, F., Stark, O. and Hohmann, S. (2019). Solution of time-variant fractional differential equations with a generalized Peano-Baker series, IEEE Control Systems Letters 3(1): 79–84.
• [14] Kaczorek, T. (2008). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223–228, DOI: 10.2478/v10006-008-0020-0.
• [15] Kaczorek, T. and Idczak, D. (2017). Cauchy formula for the time-varying linear systems with Caputo derivative, Fractional Calculus and Applied Analysis 20(2): 494–505.
• [16] Kaczorek, T. and Rogowski, K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham.
• [17] Kamocki, R. (2014). Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences 37(11): 1668–1686.
• [18] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, Amsterdam.
• [19] Li, Y., Chen, Y. and Podlubny, I. (2010). Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers and Mathematics with Applications 59(5): 1810–1821.
• [20] Lorenzo, C.F. and Hartley, T.T. (2000). Initialized fractional calculus, International Journal of Applied Mathematics 3(3): 249–265.
• [21] Luchko, Y. (2009). Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications 351(1): 218–223.
• [22] Malesza, W., Macias, M. and Sierociuk, D. (2019). Analytical solution of fractional variable order differential equations, Journal of Computational and Applied Mathematics 348: 214–236.
• [23] Martínez, L., Rosales, J., Carreño, C. and Lozano, J. (2018). Electrical circuits described by fractional conformable derivative, International Journal of Circuit Theory and Applications 46(5): 1091–1100.
• [24] Matychyn, I. (2019). Analytical solution of linear fractional systems with variable coefficients involving Riemann-Liouville and Caputo derivatives, Symmetry 11(11): 1366.
• [25] Matychyn, I. and Onyshchenko, V. (2015). Time-optimal control of fractional-order linear systems, Fractional Calculus and Applied Analysis 18(3): 687–696.
• [26] Matychyn, I. and Onyshchenko, V. (2018a). On time-optimal control of fractional-order systems, Journal of Computational and Applied Mathematics 339: 245–257.
• [27] Matychyn, I. and Onyshchenko, V. (2018b). Optimal control of linear systems with fractional derivatives, Fractional Calculus and Applied Analysis 21(1): 134–150.
• [28] Matychyn, I. and Onyshchenko, V. (2019). Optimal control of linear systems of arbitrary fractional order, Fractional Calculus and Applied Analysis 22(1): 170–179.
• [29] Matychyn, I. and Onyshchenko, V. (2020). Solution of linear fractional order systems with variable coefficients, Fractional Calculus and Applied Analysis 23(3): 753–763.
• [30] Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego.
• [31] Rockafellar, R.T. (1970). Convex Analysis, Princeton University Press, Princeton.
• [32] Si, X., Yang, H. and Ivanov, I.G. (2021). Conditions and a computation method of the constrained regulation problem for a class of fractional-order nonlinear continuous-time systems, International Journal of Applied Mathematics and Computer Science 31(1): 17–28, DOI: 10.34768/amcs-2021-0002.
• [33] Sierociuk, D. and Dzieliński, A. (2006). Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation, International Journal of Applied Mathematics and Computer Science 16(1): 129–140.
• [34] Skovranek, T., Macias, M., Sierociuk, D., Malesza, W., Dzielinski, A., Podlubny, I., Pocsova, J. and Petras, I. (2019). Anomalous diffusion modeling using ultracapacitors in domino ladder circuit, Microelectronics Journal 84: 136–141.
• [35] Zorich, V.A. and Paniagua, O. (2016). Mathematical Analysis II, Springer, Berlin/Heidelberg.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).