Necessary optimality conditions for infinite horizon variational problems on time scales

Monika Dryl; Delfim F. M. Torres

doi:10.3934/naco.2013.3.145

Article Contents

2013, Volume 3, Issue 1: 145-160. Doi: 10.3934/naco.2013.3.145

This issue Previous Article Sensitivity based trajectory following control damping methods Next Article Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory

Necessary optimality conditions for infinite horizon variational problems on time scales

Monika Dryl^1, and
Delfim F. M. Torres^2,

1.
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro
2.
CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro

Received: December 2011

Revised: November 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

We prove Euler--Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and its nabla derivative, as well as a nabla indefinite integral that depends on the unknown function.

Keywords:

Mathematics Subject Classification: Primary: 49K05; Secondary: 34N05.

Citation:

Related Papers

Cited by

References

[1]	M. Bohner and A. Peterson, "Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2001.doi: 10.1007/978-1-4612-0201-1.
[2]	M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2003.doi: 10.1007/978-0-8176-8230-9.
[3]	M. C. Caputo, Time scales: from nabla calculus to delta calculus and vice versa via duality, Int. J. Difference Equ., 5 (2010), 25-40.
[4]	S. Lang, "Undergraduate Analysis," 2^nd edition, Undergraduate Texts in Mathematics, Springer, New York, 1997.
[5]	G. Leitmann, "The Calculus of Variations and Optimal Control," Mathematical Concepts and Methods in Science and Engineering, 24, Plenum, New York, 1981.
[6]	A. B. Malinowska, N. Martins and D. F. M. Torres, Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett., 5 (2011), 41-53.doi: 10.1007/s11590-010-0189-7.
[7]	A. B. Malinowska and D. F. M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition, Proc. Est. Acad. Sci., 58 (2009), 205-212.doi: 10.3176/proc.2009.4.02.
[8]	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., 217 (2010), 1158-1162.doi: 10.1016/j.amc.2010.01.015.
[9]	A. B. Malinowska and D. F. M. Torres, A general backwards calculus of variations via duality, Optim. Lett., 5 (2011), 587-599.doi: 10.1007/s11590-010-0222-x.
[10]	N. Martins and D. F. M. Torres, Noether's symmetry theorem for nabla problems of the calculus of variations, Appl. Math. Lett., 23 (2010), 1432-1438.doi: 10.1016/j.aml.2010.07.013.
[11]	N. Martins and D. F. M. Torres, Generalizing the variational theory on time scales to include the delta indefinite integral, Comput. Math. Appl., 61 (2011), 2424-2435.doi: 10.1016/j.camwa.2011.02.022.
[12]	N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus, Comput. Math. Appl., 64 (2012), 2166-2175.doi: 10.1016/j.camwa.2011.12.006.
[13]	D. F. M. Torres, The variational calculus on time scales, Int. J. Simul. Multidisci. Des. Optim., 4 (2010), 11-25.doi: 10.1051/ijsmdo/2010003.