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On the existence of optimal solutions to fractional optimal control problems. (English) Zbl 1334.49010
Summary: We study control systems containing a fractional Riemann-Liouville derivative. The main result is a theorem on the existence of optimal solutions for such problems with a nonlinear integral performance index.

MSC:
49J21 Existence theories for optimal control problems involving relations other than differential equations
35R11 Fractional partial differential equations
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[1] Agrawal, O. P., A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38, 1, 323-337, (2004) · Zbl 1121.70019
[2] Almeida, R.; Pooseh, S.; Torres, D. F.M., Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75, 3, 1009-1025, (2012) · Zbl 1236.49042
[3] Almeida, R.; Torres, D. F.M., Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22, 12, 1816-1820, (2009) · Zbl 1183.26005
[4] Almeida, R.; Torres, D. F.M., Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16, 3, 1490-1500, (2011) · Zbl 1221.49038
[5] Bourdin, L.; Odzijewicz, T.; Torres, D. F., Existence of minimizers for fractional variational problems containing Caputo derivatives, Adv. Dyn. Syst. Appl., 8, 1, 3-12, (2013)
[6] Carpinteri, A.; Mainardi, F., Fractals and fractional calculus in continuum mechanics, (1997), Springer Berlin · Zbl 0917.73004
[7] Cesari, L., Optimization - theory and applications, (1983), Springer New York
[8] Cresson, J., Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48, 3, 033504, (2007) · Zbl 1137.37322
[9] Frederico, G. S.F.; Torres, D. F.M., A formulation of noether’s theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334, 2, 834-846, (2007) · Zbl 1119.49035
[10] Frederico, G. S.F.; Torres, D. F.M., Fractional conservation laws in optimal control theory, Nonlinear Dyn., 53, 3, 215-222, (2008) · Zbl 1170.49017
[11] Frederico, G. S.F.; Torres, D. F.M., Fractional noether’s theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217, 3, 1023-1033, (2010) · Zbl 1200.49019
[12] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45, 765-771, (2006)
[13] Hilfer, R., Applications of fractional calculus in physics, (2000), World Sci. Publishing River Edge, NJ · Zbl 0998.26002
[14] Idczak, D.; Kamocki, R., On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in \(\mathbb{R}^n\), Fractional Calculus Appl. Anal., 14, 4, 538-553, (2011) · Zbl 1273.34010
[15] Idczak, D.; Majewski, M., Fractional fundamental lemma of order \(\alpha \in(n - \frac{1}{2}, n)\) with \(n \in \mathbb{N}\), \(n \geqslant 2\), Dyn. Syst. Appl., 21, 2-3, 251268, (2012)
[16] Jelicic, Z. D.; Petrovacki, N., Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38, 571-581, (2009) · Zbl 1274.49035
[17] Kamocki, R., Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., (2013)
[18] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003
[19] Kisielewicz, M., Differential inclusions and optimal control, (1991), PWN Warsaw · Zbl 0731.49001
[20] Klimek, M., On solutions of linear fractional differential equations of a variational type, (2009), The Publishing Office of Czestochowa University of Technology Czestochowa
[21] Łojasiewicz, S., An introduction to the theory of real functions, (1973), PWN Warsaw (in Polish)
[22] Malinowska, A. B., A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25, 11, 1941-1946, (2012) · Zbl 1259.49005
[23] Malinowska, A. B.; Torres, D. F.M., Introduction to the fractional calculus of variations, (2012), Imperial College Press London · Zbl 1258.49001
[24] Malinowska, A. B.; Torres, D. F.M., Multiobjective fractional variational calculus in terms of a combined Caputo derivative, Appl. Math. Comput., 218, 9, 5099-5111, (2012) · Zbl 1238.49029
[25] M.W. Michalski, Derivatives of noninteger order and their applications, Dissertationes mathematicae, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 1993.
[26] Mozyrska, D.; Torres, D. F.M., Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Process., 91, 3, 379-385, (2011) · Zbl 1203.94046
[27] Musielak, J., An introduction to the functional analysis, (1989), PWN Warsaw (in Polish)
[28] Olech, C., A characterization of \(L^1\)-weak lower semicontinuity of integral functionals, Bull. Acad. Pol. Sci., 25, 135-142, (1977) · Zbl 0395.46026
[29] Pooseh, S.; Almeida, R.; Torres, D. F.M., Fractional order optimal control problems with free terminal time, J. Ind. Manage. Optim., 10, 2, 363-381, (2014) · Zbl 1278.26013
[30] Riewe, F., Mechanics with fractional derivatives, Phys. Rev. E (3), 55, 3 part B, 3581-3592, (1997)
[31] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives and some their applications (in Russian), (1987), Nauka i Technika Minsk, (see also the english edition: S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives - Theory and Applications. Gordon and Breach: Amsterdam, 1993.) · Zbl 0617.26004
[32] Tricaud, Ch.; Chen, Y., Time-optimal control of systems with fractional dynamics, Int. J. Differ. Equ., (2010), (Article ID 461048, 16 p.) · Zbl 1203.49031
[33] West, B. J.; Grigolini, P., Applications of fractional calculus in physics, (1998), World Scientific Singapore
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