Generalized transversality conditions for the Hahn quantum variational calculus. (English) Zbl 1262.39011
Summary: We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory.
MSC:
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 39A70 | Difference operators |
| 49J05 | Existence theories for free problems in one independent variable |
| 49K05 | Optimality conditions for free problems in one independent variable |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
Keywords:
Hahn’s difference operator; Jackson-Norlünd’s integral; quantum calculus; calculus of variations; Euler-Lagrange equation; generalized natural boundary conditionsReferences:
| [1] | Aldwoah KA, Int. J. Math. Stat. 9 pp 106– (2011) |
| [2] | DOI: 10.1177/1077546309103268 · Zbl 1273.49024 · doi:10.1177/1077546309103268 |
| [3] | DOI: 10.1016/j.jmaa.2009.06.029 · Zbl 1169.49016 · doi:10.1016/j.jmaa.2009.06.029 |
| [4] | DOI: 10.1016/j.cam.2005.06.046 · Zbl 1108.33008 · doi:10.1016/j.cam.2005.06.046 |
| [5] | Atici FM, Nonlinear Dyn. Syst. Theory 9 pp 1– (2009) |
| [6] | DOI: 10.1016/j.aml.2007.03.013 · Zbl 1153.90001 · doi:10.1016/j.aml.2007.03.013 |
| [7] | Aulbach B, A unified approach to continuous and discrete dynamics, in Qualitative Theory of Differential Equations B. Sz-Nagy and L. Hatvani, eds., North-Holland (1990) |
| [8] | DOI: 10.3166/ejc.17.9-18 · Zbl 1248.49024 · doi:10.3166/ejc.17.9-18 |
| [9] | Brito da Cruz AMC, Nonlinear Anal., Theory Methods Appl., Ser. A |
| [10] | Caputo MR, Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications (2005) · Zbl 1092.91061 · doi:10.1017/CBO9780511806827 |
| [11] | DOI: 10.1504/IJMMNO.2010.031750 · Zbl 1218.49038 · doi:10.1504/IJMMNO.2010.031750 |
| [12] | DOI: 10.1016/j.cam.2005.06.009 · Zbl 1119.39017 · doi:10.1016/j.cam.2005.06.009 |
| [13] | DOI: 10.3176/proc.2008.2.03 · Zbl 1161.33302 · doi:10.3176/proc.2008.2.03 |
| [14] | DOI: 10.1016/j.aml.2010.08.023 · Zbl 1208.49020 · doi:10.1016/j.aml.2010.08.023 |
| [15] | DOI: 10.1007/978-3-540-69532-5_9 · Zbl 1191.49017 · doi:10.1007/978-3-540-69532-5_9 |
| [16] | DOI: 10.1002/mana.19490020103 · Zbl 0031.39001 · doi:10.1002/mana.19490020103 |
| [17] | DOI: 10.1007/978-1-4613-0071-7 · doi:10.1007/978-1-4613-0071-7 |
| [18] | DOI: 10.1007/s11590-010-0189-7 · Zbl 1233.90265 · doi:10.1007/s11590-010-0189-7 |
| [19] | DOI: 10.1002/mma.1289 · Zbl 1196.49014 · doi:10.1002/mma.1289 |
| [20] | DOI: 10.1007/s10957-010-9730-1 · Zbl 1218.49026 · doi:10.1007/s10957-010-9730-1 |
| [21] | Malinowska AB, Optim. Lett. 4 (2010) |
| [22] | Malinowska AB, Z. Naturforsch. A 66 (2011) · doi:10.1515/zna-2011-6-704 |
| [23] | DOI: 10.1016/j.na.2008.11.035 · Zbl 1238.49037 · doi:10.1016/j.na.2008.11.035 |
| [24] | DOI: 10.1016/j.aml.2010.07.013 · Zbl 1205.49033 · doi:10.1016/j.aml.2010.07.013 |
| [25] | DOI: 10.1016/j.camwa.2011.02.022 · Zbl 1221.49040 · doi:10.1016/j.camwa.2011.02.022 |
| [26] | DOI: 10.1016/j.cam.2006.05.005 · Zbl 1119.33010 · doi:10.1016/j.cam.2006.05.005 |
| [27] | DOI: 10.1080/00207729708929447 · Zbl 0878.90012 · doi:10.1080/00207729708929447 |
| [28] | van Brunt B, The Calculus of Variations (2004) · Zbl 1039.49001 · doi:10.1007/b97436 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
