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Generalizing the variational theory on time scales to include the delta indefinite integral. (English) Zbl 1221.49040
Summary: We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta derivative, but also on a delta indefinite integral that depends on the unknown function. Such kinds of variational problems were considered by Euler himself and have been recently investigated by J. Gregory [Methods Appl. Anal. 15, No. 4, 427–436 (2008; Zbl 1178.49024)]. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases.
MSC:
49K21Optimal control problems involving relations other than differential equations
34A08Fractional differential equations
References:
[1]Bohner, M.; Peterson, A.: Dynamic equations on time scales, (2001)
[2]Bohner, M.; Peterson, A.: Advances in dynamic equations on time scales, (2003)
[3]Bohner, M.: Calculus of variations on time scales, Dyn. syst. Appl. 13, No. 3–4, 339-349 (2004) · Zbl 1069.39019
[4]Bartosiewicz, Z.; Martins, N.; Torres, D. F. M.: The second Euler–Lagrange equation of variational calculus on time scales, Eur. J. Control 17, No. 1, 9-18 (2011)
[5]Bohner, M.; Ferreira, R. A. C.; Torres, D. F. M.: Integral inequalities and their applications to the calculus of variations on time scales, Math. inequal. Appl. 13, No. 3, 511-522 (2010) · Zbl 1190.26015 · doi:http://files.ele-math.com/abstracts/mia-13-35-abs.pdf
[6]E. Girejko, A.B. Malinowska, D.F.M. Torres, The contingent epiderivative and the calculus of variations on time scales, Optimization (2010), in press (doi:10.1080/02331934.2010.506615).
[7]Malinowska, A. B.; Martins, N.; Torres, D. F. M.: Transversality conditions for infinite horizon variational problems on time scales, Optim. lett. 5, No. 1, 41-53 (2011) · Zbl 1233.90265 · doi:10.1007/s11590-010-0189-7
[8]Malinowska, A. B.; Torres, D. F. M.: Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales, Appl. math. Comput. 217, No. 3, 1158-1162 (2010) · Zbl 1200.49022 · doi:10.1016/j.amc.2010.01.015
[9]Fraser, C. G.: Isoperimetric problems in the variational calculus of Euler and Lagrange, Historia math. 19, No. 1, 4-23 (1992) · Zbl 0743.01015 · doi:10.1016/0315-0860(92)90052-D
[10]Gregory, J.: Generalizing variational theory to include the indefinite integral, higher derivatives, and a variety of means as cost variables, Methods appl. Anal. 15, No. 4, 427-435 (2008) · Zbl 1178.49024 · doi:euclid:maa/1254492827
[11]Caputo, M. C.: Time scales: from nabla calculus to delta calculus and vice versa via duality, Int. J. Differ. equ. 5, No. 1, 25-40 (2010)
[12]A.B. Malinowska, D.F.M. Torres, A general backwards calculus of variations via duality, Optim. Lett. (2010), in press (doi:10.1007/s11590-010-0222-x).
[13]Martins, N.; Torres, D. F. M.: Calculus of variations on time scales with nabla derivatives, Nonlinear anal. 71, No. 12, e763-e773 (2009)
[14]Van Brunt, B.: The calculus of variations, (2004)
[15]Ferreira, R. A. C.; Torres, D. F. M.: Isoperimetric problems of the calculus of variations on time scales, Contemporary mathematics 514, 123-131 (2010) · Zbl 1222.49031
[16]Kac, V.; Cheung, P.: Quantum calculus, (2002)
[17]Malinowska, A. B.; Torres, D. F. M.: The Hahn quantum variational calculus, J. optim. Theory appl. 147, No. 3, 419-442 (2010) · Zbl 1218.49026 · doi:10.1007/s10957-010-9730-1
[18]A.M.C. Brito da Cruz, N. Martins, D.F.M. Torres, Higher-order Hahn’s quantum variational calculus, Nonlinear Anal. (2011), in press (doi:10.1016/j.na.2011.01.015).
[19]Pawłuszewicz, E.; Torres, D. F. M.: Backward linear control systems on time scales, Internat. J. Control 83, No. 8, 1573-1580 (2010) · Zbl 1200.93100 · doi:10.1080/00207179.2010.483562
[20]Martins, N.; Torres, D. F. M.: Noether’s symmetry theorem for nabla problems of the calculus of variations, Appl. math. Lett. 23, No. 12, 1432-1438 (2010) · Zbl 1205.49033 · doi:10.1016/j.aml.2010.07.013
[21]Almeida, R.; Torres, D. F. M.: Isoperimetric problems on time scales with nabla derivatives, J. vib. Control 15, No. 6, 951-958 (2009)
[22]Atici, F. M.; Biles, D. C.; Lebedinsky, A.: An application of time scales to economics, Math. comput. Modelling 43, No. 7-8, 718-726 (2006) · Zbl 1187.91125 · doi:10.1016/j.mcm.2005.08.014
[23]Caputo, M. R.: A unified view of ostensibly disparate isoperimetric variational problems, Appl. math. Lett. 22, No. 3, 332-335 (2009) · Zbl 1155.49003 · doi:10.1016/j.aml.2008.04.004
[24]Agarwal, R. P.; Bohner, M.; Wong, P. J. Y.: Sturm–Liouville eigenvalue problems on time scales, Appl. math. Comput. 99, No. 2–3, 153-166 (1999) · Zbl 0938.34015 · doi:10.1016/S0096-3003(98)00004-6