Partial Derivatives and Optimization Problem on Time Scales

Authors: Francisco Miranda

Abstract:

The optimization problem using time scales is studied. Time scale is a model of time. The language of time scales seems to be an ideal tool to unify the continuous-time and the discrete-time theories. In this work we present necessary conditions for a solution of an optimization problem on time scales. To obtain that result we use properties and results of the partial diamond-alpha derivatives for continuous-multivariable functions. These results are also presented here.

Keywords: Lagrange multipliers, mathematical programming, optimization problem, time scales.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331285

References:

[1] B. Aulbach and S. Hilger, "A unified approach to continuous and discrete dynamics," in Qualitative theory of differential equations (Szeged, 1988), vol. 53 of Colloq. Math. Soc. J'anos Bolyai, North-Holland, Amsterdam, 1990, pp. 37-56.
[2] S. Hilger, "Ein maßkettenkalk┬¿ul mit anwendung auf zentrumsmannigfaltigkeiten," Ph.D. dissertation, Universit┬¿at W┬¿urzburg, Germany, 1988.
[3] B. Jackson, "Partial dynamic equations on time scales," J. Comput. Appl. Math., vol. 186, no. 2, pp. 391-415, 2006.
[4] M. Bohner and G. SH. Guseinov, "Partial differentiation on time scales," Dynam. Systems Appl., vol. 13, pp. 351-379, 2004.
[5] M. Z. Sarikaya, N. Aktan, H. Yildirim, and K. İIlarslan, "Partial Δ- differentiation for multivariable functions on n-dimensional time scales," J. Math. Inequal., vol. 3, no. 2, pp. 277-291, 2009.
[6] P. Stehl'─▒k, "Maximum principles for elliptic dynamic equations," Math. Comput. Model., vol. 51, no. 9-10, pp. 1193-1201, 2010.
[7] U. M. O¨ zkan and B. Kaymakalan, "Basics of diamond-α partial dynamic calculus on time scales," Math. Comput. Model., vol. 50, no. 9-10, pp. 1253-1261, 2009.
[8] A. B. Malinowska and D. F. M. Torres, "Strong minimizers of the calculus of variations on time scales and the Weierstrass condition," in Proc. Estonian Acad. Sci. Phys. Math., vol. 58, no. 4, 2009, pp. 205-212.
[9] A. B. Malinowska and D. F. M. Torres, "The delta-nabla calculus of variations," Fasc. Math., vol. 44, pp. 75-83, 2010.
[10] A. B. Malinowska and D. F. M. Torres, "Leitmann-s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales," Appl. Math. Comput., vol. 217, no. 3, pp. 1158-1162, 2010.
[11] M. Bohner, "Calculus of variations on time scales," Dynam. Systems Appl., vol. 13, no. 3-4, pp. 339-349, 2004.
[12] N. Martins and D. F. M. Torres, "Calculus of variations on time scales with nabla derivatives," Nonlinear Anal., vol. 71, no. 12, pp. e763-e773, 2009.
[13] R. A. C. Ferreira, A. B. Malinowska, and D. F. M. Torres, "Optimality conditions for the calculus of variations with higher-order delta derivatives," Appl. Math. Lett., vol. 24, no. 1, pp. 87-92, 2011.
[14] R. Hilscher, "Optimality conditions for time scale variational problems," D.Sc. dissertation, Masaryk University, Brno, Duben, 2008.
[15] M. Adivar and S.-C. Fang, "Convex optimization on mixed domains," J. Ind. Manag. Optim., vol. 8, no. 1, pp. 189-227, 2012.
[16] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications. Boston, MA: Birkh¨auser Boston, Inc., 2001.
[17] M. Bohner and A. Peterson (Eds.), Advances in Dynamic Equations on Time Scales. Boston, MA: Birkh¨auser Boston, Inc., 2003.
[18] J. W. Rogers Jr. and Q. Sheng, "Notes on the diamond-╬▒ dynamic derivative on time scales," J. Math. Anal., vol. 326, no. 1, pp. 228-241, 2007.
[19] R. K. Sundaram, A First Course in Optimization Theory. Cambridge, UK: Cambridge University Press, 1996.
[20] A. B. Malinowska and D. F. M. Torres, "The diamond-alpha Riemann integral and mean value theorems on time scales," Dynam. Systems Appl., vol. 18, pp. 469-482, 2009.