zbMATH — the first resource for mathematics

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.

49J21 Existence theories for optimal control problems involving relations other than differential equations
35R11 Fractional partial differential equations
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.