We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions.
1. Almeida, R. and Torres, D. F. M. Isoperimetric problems on time scales with nabla derivatives. J. Vib. Control, 2009, 15, 951–958.
doi:10.1177/1077546309103268
2. Atici, F. M., Biles, D. C., and Lebedinsky, A. An application of time scales to economics. Math. Comput. Model., 2006, 43, 718–726.
doi:10.1016/j.mcm.2005.08.014
3. Atici, F. M. and Uysal, F. A production-inventory model of HMMS on time scales. Appl. Math. Lett., 2008, 21, 236–243.
doi:10.1016/j.aml.2007.03.013
4. Bangerezako, G. Variational q-calculus. J. Math. Anal. Appl., 2004, 289, 650–665.
doi:10.1016/j.jmaa.2003.09.004
5. Bartosiewicz, Z. and Torres, D. F. M. Noether’s theorem on time scales. J. Math. Anal. Appl., 2008, 342, 1220–1226.
doi:10.1016/j.jmaa.2008.01.018
6. Bohner, M. Calculus of variations on time scales. Dynam. Syst. Appl., 2004, 13, 339–349.
7. Bohner, M. and Guseinov, G. Sh. Riemann and Lebesgue integration. In Advances in Dynamic Equations on Time Scales (Bohner, M. and Peterson, A. C., eds). Birkhäuser Boston, Boston, MA, 2003, 117–163.
8. Bohner, M. and Guseinov, G. Sh. Partial differentiation on time scales. Dynam. Syst. Appl., 2004, 13, 351–379.
9. Bohner, M. and Guseinov, G. Sh. Double integral calculus of variations on time scales. Comput. Math. Appl., 2007, 54, 45–57.
doi:10.1016/j.camwa.2006.10.032
10. Bohner, M. and Peterson, A. (eds) Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Boston, MA, 2001.
11. Bohner, M. and Peterson, A. (eds) Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Boston, MA, 2003.
12. Brechtken-Manderscheid, U. Introduction to the Calculus of Variations. Translated from the German by P. G. Engstrom. Chapman & Hall, London, 1991.
13. Ferreira, R. A. C. and Torres, D. F. M. Remarks on the calculus of variations on time scales. Int. J. Ecol. Econ. Stat., 2007, 9, 65–73.
14. Ferreira, R. A. C. and Torres, D. F. M. Higher-order calculus of variations on time scales. In Mathematical Control Theory and Finance. Springer, Berlin, 2008, 149–159.
15. Hilscher, R. and Zeidan, V. Calculus of variations on time scales: weak local piecewise C1rd solutions with variable endpoints. J. Math. Anal. Appl., 2004, 289, 143–166.
doi:10.1016/j.jmaa.2003.09.031
16. Hilscher, R. and Zeidan, V. Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal., 2009, 70, 3209–3226.
doi:10.1016/j.na.2008.04.025
17. Kac, V. and Cheung, P. Quantum Calculus. Springer, New York, 2002.
18. Kelley, W. G. and Peterson, A. C. Difference Equations. Academic Press, Boston, MA, 1991.
19. Leitmann, G. The Calculus of Variations and Optimal Control. Plenum, New York, 1981.
20. Malinowska, A. B. and Torres, D. F. M. Necessary and sufficient conditions for local Pareto optimality on time scales. J. Math. Sci. (N. Y.), 2009, in press.
doi:10.1007/s10958-009-9601-1
21. Martins, N. and Torres, D. F. M. Calculus of variations on time scales with nabla derivatives. arXiv:0807.2596v1[math.OC]
doi: 10.1016/j.na.2008.11.035.
22. Seiffertt, J., Sanyal, S., and Wunsch, D. C. Hamilton–Jacobi–Bellman equations and approximate dynamic programming on time scales. IEEE Trans. Syst. Man Cybern., Part B: Cybern., 2008, 38, 918–923.
doi:10.1109/TSMCB.2008.923532
