The $l$-moment problem and optimal control for systems modeled by fractional equations with multiparameter and ``nonsingular'' derivatives

S. S. Postnov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 20 International Saratov Winter School "Contemporary Problems of Function Theory and Their Applications", Saratov, January 28 — February 1, 2020. Part 1, Tome 199 (2021), pp. 86-116.

Résumé

This paper devoted to the problem of non-uniqueness of the definition of fractional operators and the influence of this problem on the properties of solutions of optimal-control problems for fractional systems. We consider two optimal-control problems for systems with lumped parameters whose dynamics is described by linear fractional integro-differential equations: the problem of search for a control with minimal norm at given control time and the time-optimal problem at given restriction on the norm of the control. We analyze the problems with the fractional derivative operators in the sense of Erdélyi–Kober, Katugampola, Atangana–Baleanu, and Caputo–Fabrizio. These problems are reduced to the $l$-moment problem and the well-posedness and solvability conditions are obtained. Several examples of the explicit calculation of optimal controls are given and the properties of the solutions obtained that depend on the type of the fractional derivative operator are compared.

Détail

Keywords: fractional-order dynamic systems, optimal control, fractional derivatives, $l$-problem of moments.

Bibliographie

[1] Butkovskii A. G., Metody upravleniya sistemami s raspredelennymi parametrami, Nauka, M., 1975

[2] Butkovskii A. G., Postnov S. S., Postnova E. A., “Drobnoe integro-differentsialnoe ischislenie i ego prilozheniya v teorii upravleniya. II. Drobnye dinamicheskie sistemy: modelirovanie i apparatnaya realizatsiya”, Avtomat. telemekh., 2013, no. 5, 3–34 | Zbl

[3] Zvyagin V. G., Orlov V. P., “O slaboi razreshimosti zadachi vyazkouprugosti s pamyatyu”, Differ. uravn., 53:2 (2017), 215–220 | Zbl

[4] Kubyshkin V. A., Postnov S. S., “Zadacha optimalnogo upravleniya lineinoi statsionarnoi sistemoi drobnogo poryadka v forme problemy momentov: postanovka i issledovanie”, Avtomat. telemekh., 2014, no. 5, 3–17 | Zbl

[5] Morozov A. N., Skripkin A. V., Nemarkovskie fizicheskie protsessy, Fizmatlit, M., 2018

[6] Postnov S. S., “Zadachi optimalnogo upravleniya dlya lineinykh sistem drobnogo poryadka, zadannykh uravneniyami s proizvodnoi Adamara”, Dokl. RAN., 476:2 (2017), 143–147 | Zbl

[7] Postnov S. S., “Zadachi optimalnogo upravleniya dlya nekotorykh lineinykh sistem drobnogo poryadka, zadannykh uravneniyami s proizvodnoi Khilfera”, Probl. upravl., 2018, no. 5, 14–25

[8] Postnov S. S., “Zadachi optimalnogo upravleniya dlya sistem, modeliruemykh uravneniyami drobnogo poryadka s mnogoparametricheskimi proizvodnymi”, Mat. 20 Mezhdunar. Saratov. zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 28 yanvarya — 1 fevralya 2020 g.), Nauchnaya kniga, Saratov, 2020, 335–339

[9] Postnov S. S., “Issledovanie zadachi optimalnogo upravleniya dlya odinochnogo i dvoinogo integratorov drobnogo poryadka s pomoschyu metoda momentov”, Probl. upravl., 2012, no. 5, 9–17

[10] Postnov S. S., “$l$-Problema momentov dlya odnomernykh lineinykh integro-differentsialnykh uravnenii s operatorami Erdeii—Kobera”, Dokl. RAN., 486:6 (2019), 659–662 | Zbl

[11] Rudenko O. V., “Nelineinye integro-differentsialnye modeli dlya intensivnykh voln v sredakh tipa biotkanei i geterostruktur so slozhnoi vnutrennei dinamikoi relaksatsionnogo tipa”, Akust. zh., 60:4 (2014), 368–375

[12] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987

[13] Uchaikin V. V., Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008

[14] Agrawal O. P., “A general formulation and solution scheme for fractional optimal control problems”, Nonlin. Dyn., 38 (2004), 323-–337 | DOI | Zbl

[15] Al-Saqabi B., Kiryakova V., “Explicit solutions of fractional integral and differential equations involving Erdélyi–Kober operators”, Appl. Math. Comput., 95 (1998), 1-–13 | DOI | MR | Zbl

[16] Ammi M. R. S., Torres D. F. M., “Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives”, Comput. Math. Appl., 78:5 (2019), 1507–1516 | DOI | MR | Zbl

[17] Atangana A., Baleanu D., “New fractional derivatives with nonlocal and nonsingular kernel”, Thermal Sci., 20:2 (2016), 763–769 | DOI

[18] Atangana A., “On the new fractional derivative and application to nonlinear Fisher's reaction–diffusion equation”, Appl. Math. Comput., 273 (2016), 948–-956 | MR | Zbl

[19] Baleanu D., “On fractional operators and their classifications”, Mathematics., 7 (2019), 830 | DOI

[20] Baleanu D., Fernandez A., “On some new properties of fractional derivatives with Mittag-Leffler kernel”, Commun. Nonlin. Sci. Num. Simul., 59 (2018), 444–-462 | DOI | Zbl

[21] Baleanu D., Wu G.–C., Zeng S.–D., “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations”, Chaos Solitons Fractals., 102 (2017), 99–-105 | DOI | MR | Zbl

[22] Bhairat S. P., Dhaigude D.-B., “Existence of solutions of generalized fractional differential equation with nonlocal initial condition”, Math. Bohem., 144:2 (2019), 203-–220 | DOI | MR | Zbl

[23] Caputo M., Fabrizio M., “A new definition of fractional derivative without singular kernel”, Progr. Fract. Differ. Appl., 1:2 (2015), 73–85

[24] Concezzi M., Garra R., Spigler R., “Fractional relaxation and fractional oscillation models involving Erdélyi–Kober integrals”, Fract. Calc. Appl. Anal., 18:5 (2015), 1212–-1231 | DOI | MR | Zbl

[25] ElNabulsi R. A., “The fractional calculus of variations from extended Erdélyi–Kober operator”, Int. J. Modern Phys. B., 23:16 (2009), 3349–3361 | DOI

[26] Fernandez A., Özarslan M. A., Baleanu D., “On fractional calculus with general analytic kernels”, Appl. Math. Comput., 354 (2019), 248–265 | MR | Zbl

[27] Gambo Y. Y., Jarad F., Baleanu D., Abdeljawad T., “On Caputo modification of the Hadamard fractional derivatives”, Adv. Differ. Equations., 2014 (2014), 10 | DOI | MR | Zbl

[28] Garra R., Mainardi F., Spada G., “A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus”, Chaos Solitons Fractals., 102 (2017), 333–338 | DOI | MR

[29] Hilfer R., “Experimental evidence for fractional time evolution in glass forming materials”, Chem. Phys., 284 (2002), 399–-408 | DOI

[30] Hilfer R., “Fractional time evolution”, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000, 87–-130 | DOI | Zbl

[31] Jarad F., Abdeljawad T., Hammouch Z., “On a class of ordinary differential equations in the frame of Atangana–-Baleanu fractional derivative”, Chaos Solitons Fractals., 117 (2018), 16–20 | DOI | MR | Zbl

[32] Kamocki R., “Pontryagin maximum principle for fractional ordinary optimal control problems”, Math. Meth. Appl. Sci., 37:11 (2014), 1668-–1686 | DOI | Zbl

[33] Kamocki R., Majewski M., “Fractional linear control systems with Caputo derivative and their optimization”, Optimal Control Appl. Meth., 36:6 (2015), 953–967 | DOI | MR | Zbl

[34] Katugampola U. N., “A new approach to generalized fractional derivatives”, Bull. Math. Anal. Appl., 6:4 (2014), 1–15 | MR | Zbl

[35] Katugampola U. N., Existence and uniqueness results for a class of generalized fractional differential equations, arXiv: 1411.5229v2 [math.CA]

[36] Katugampola U. N., “New approach to a generalized fractional integral”, Appl. Math. Comput., 218 (2011), 860–865 | MR | Zbl

[37] Khan A., Abro K. A., Tassaddiq A., Khan I., “Atangana–-Baleanu and Caputo–Fabrizio analysis of fractional derivatives for heat and mass transfer of second grade fluids over a vertical plate: A comparative study”, Entropy., 19 (2017), 279 | DOI

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

[39] Kiryakova V., Generalized Fractional Calculus and Applications, Wiley, New York, 1994 | Zbl

[40] Kolokoltsov V. N., “The probabilistic point of view on the generalized fractional partial differential equations”, Fract. Calc. Appl. Anal., 22:3 (2019), 543–-600 | DOI | MR | Zbl

[41] Kubyshkin V. A., Postnov S. S., “The optimal control problem for linear systems of non-integer order with lumped and distributed parameters”, Discontin. Nonlin. Complexity., 4:4 (2015), 429–443 | DOI | Zbl

[42] Kubyshkin V. A., Postnov S. S., “Optimal control problem investigation for linear time-invariant systems of fractional order with lumped parameters described by equations with Riemann–Liouville derivative”, J. Control Sci. Eng., 2016 (2016), 4873083 | DOI | Zbl

[43] Lian T. T., Fan Z. B., Li G., “Time optimal controls for fractional differential systems with Riemann–Liouville derivatives”, Fract. Calc. Appl. Anal., 21:6 (2018), 1524–-1541 | DOI | MR

[44] Losada J., Nieto J. J., “Properties of a new fractional derivative without singular kernel”, Progr. Fract. Differ. Appl., 1:2 (2015), 87–-92

[45] Luchko Yu., Trujillo J. J., “Caputo-type modification of the Erdélyi–Kober fractional derivative”, Fract. Calc. Appl. Anal., 10 (2007), 249-–267 | MR | Zbl

[46] Monje C. A., Chen Y. Q., Vinagre B. M., Xue D., Feliu V., Fractional-order systems and controls: Fundamentals and applications, Springer-Verlag, London, 2010 | Zbl

[47] Mozyrska D., Torres D. F. M., “Modified optimal energy and initial memory of fractional continuous-time linear systems”, Signal Process., 91:3 (2011), 379–-385 | DOI | Zbl

[48] Postnov S., “Optimal control problem for linear fractional-order systems described by equations with Hadamard-type derivative”, J. Phys. Conf. Ser., 918:1 (2017), 012026 | DOI

[49] Riaz M. B., Zafar A. A., “Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo–Fabrizio derivatives”, Math. Model. Nat. Phenom., 13 (2018), 8 | DOI | Zbl

[50] Saxena R. K., Mathai A. M., Haubold H. J., “Computational solutions of unified fractional reaction–diffusion equations with composite fractional time derivative”, Commun. Nonlin. Sci. Num. Simul., 27 (2015), 1–-11 | DOI | Zbl

[51] Tarasov V. E., Fractional Dynamics, Springer-Verlag, Berlin, 2010 | Zbl

[52] Sales Teodoro G., Tenreiro Machado J. A., Capelas de Oliveira E., “A review of definitions of fractional derivatives and other operators”, J. Comp. Phys., 388 (2019), 195-–208 | DOI | Zbl

[53] Tricaud C., Chen Y. Q., “Time-optimal control of systems with fractional dynamics”, Int. J. Differ. Equations., 2010 (2010), 461048 | DOI | MR | Zbl

[54] Vivek D., Kanagarajan K., Sivasundaram S., “Dynamics and stability results for Hilfer fractional-type thermistor problem”, Fractal Fract., 1 (2017), 5 | DOI | Zbl

[55] Yang X. J., Srivastava H. M., Tenreiro Machado J. A., “A new fractional derivative without singular kernel. Application to the modelling of the steady heat flow”, Therm. Sci., 20 (2016), 753–-756 | DOI

[56] Zhang S., Hu L., Sun S., “The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo–Fabrizio derivative”, J. Nonlin. Sci. Appl., 11 (2018), 428–-436 | DOI | Zbl