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The necessary conditions of fractional optimal control in the sense of Caputo. (English) Zbl 1263.49018
Summary: This paper deals with the optimal control problem of a fractional dynamic system in the sense of Caputo. The main result of this paper gives a second order necessary optimality condition for fractional optimal control problems, which has not been discussed before. An application is introduced to explain our main results.

49K15 Optimality conditions for problems involving ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
Full Text: DOI
[1] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88, (1991)
[2] Glockle, W.G.; Nonnenmacher, T.F., A fractional calculus approach of selfsimilar protein dynamics, Biophys. J., 68, 46-53, (1995)
[3] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[4] Mainardi, F.; Carpinteri, A. (ed.); Mainardi, F. (ed.), Fractional calculus: some basic problems in continuum and statistical mechanics, 291-348, (1997), Wien · Zbl 0917.73004
[5] Metzler, F.; Schick, W.; Kilian, H.G.; Nonnenmacher, T.F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103, 7180-7186, (1995)
[6] Sidi Ammi, M.R.; El Kinani, E.H.; Torres, D.F.M., Existence and uniqueness of solutions to functional integro-differential fractional equations, Electron. J. Differ. Equ., 103, 1-9, (2012)
[7] Sidi Ammi, M.R.; Torres, D.F.M., Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives, Differ. Equ. Appl., 4, 267-276, (2012) · Zbl 1244.26018
[8] Agarwal, R.P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 973-1033, (2010) · Zbl 1198.26004
[9] Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338, 1340-1350, (2008) · Zbl 1209.34096
[10] Zhang, S., Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 136-148, (2003) · Zbl 1026.34008
[11] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., Real World Appl., 11, 4465-4475, (2010) · Zbl 1260.34017
[12] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear Anal., Real World Appl., 12, 262-272, (2011) · Zbl 1214.34010
[13] Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012) · Zbl 1258.49001
[14] Almeida, R.; Ferreira, R.A.C.; Torres, D.F.M., Isoperimetric problems of the calculus of variations with fractional derivatives, Acta Math. Sci., Ser. B, Engl. Ed., 32, 619-630, (2012) · Zbl 1265.49022
[15] Odzijewicz, T.; Malinowska, A.B.; Torres, D.F.M., Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75, 1507-1515, (2012) · Zbl 1236.49043
[16] Frederico, G.S.F.; Torres, D.F.M., Fractional optimal control in the sense of Caputo and the fractional noether’s theorem, Int. Math. Forum, 3, 479-493, (2008) · Zbl 1154.49016
[17] Agrawal, O.P., Fractional variational calculus and the transversality conditions, J. Phys. A, Math. Gen., 39, 10375-10384, (2006) · Zbl 1097.49021
[18] 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
[19] Ding, Y., Wang, Z., Ye, H.: Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol. 20(3) (2012)
[20] Stengel, R.F., Mutation and control of the human immunodeficiency virus, Math. Biosci., 213, 93-102, (2008) · Zbl 1139.92310
[21] Perelson, A.S.; Kirschner, D.E.; Boer, R., Dynamics of HIV infection of CD4\^{+}\(T\)-cells, Math. Biosci., 114, 81-125, (1993) · Zbl 0796.92016
[22] Brandt, M.E.; Chen, G., Feedback control of a biodynamical model of HIV-1, IEEE Trans. Biomed. Eng., 48, 754-758, (2001)
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