×

Necessary conditions for a class of optimal control problems on time scales. (English) Zbl 1163.49013

Summary: Based on the Gâteaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results.

MSC:

49J50 Fréchet and Gateaux differentiability in optimization
49K15 Optimality conditions for problems involving ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001
[2] M. Bohner, “Calculus of variations on time scales,” Dynamic Systems and Applications, vol. 13, no. 3-4, pp. 339-349, 2004. · Zbl 1069.39019
[3] R. Hilscher and V. Zeidan, “Calculus of variations on time scales: weak local piecewise Crd1 solutions with variable endpoints,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 143-166, 2004. · Zbl 1043.49004
[4] F. M. Atici, D. C. Biles, and A. Lebedinsky, “An application of time scales to economics,” Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 718-726, 2006. · Zbl 1187.91125
[5] F. M. Atici and F. Uysal, “A production-inventory model of HMMS on time scales,” Applied Mathematics Letters, vol. 21, no. 3, pp. 236-243, 2008. · Zbl 1153.90001
[6] R. A. C. Ferreira and D. F. M. Torres, “Higher-order calculus of variations on time scales,” in Mathematical Control Theory and Finance, pp. 149-159, Springer, New York, NY, USA, 2008. · Zbl 1191.49017
[7] R. Hilscher and V. Zeidan, “Weak maximum principle and accessory problem for control problems on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3209-3226, 2009. · Zbl 1157.49030
[8] A. Cabada and D. R. Vivero, “Expression of the Lebesgue \Delta -integral on time scales as a usual Lebesgue integral: application to the calculus of \Delta -antiderivatives,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 194-207, 2006. · Zbl 1092.39017
[9] G. Sh. Guseinov, “Integration on time scales,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 107-127, 2003. · Zbl 1039.26007
[10] A. Cabada and D. R. Vivero, “Criterions for absolute continuity on time scales,” Journal of Difference Equations and Applications, vol. 11, no. 11, pp. 1013-1028, 2005. · Zbl 1081.39011
[11] L. Jiang and Z. Zhou, “Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1376-1388, 2008. · Zbl 1189.34170
[12] R. A. Adams, Sobolev Spaces, vol. 6 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975. · Zbl 0314.46030
[13] N. Martins and D. F. M. Torres, “Calculus of variations on time scales with nabla derivatives,” Nonlinear Analysis: Theory, Methods & Applications. In press. · Zbl 1238.49037
[14] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001
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.