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2013, 3(1): 145-160. doi: 10.3934/naco.2013.3.145

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, Portugal
2. 

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

Received  December 2011 Revised  November 2012 Published  January 2013

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: transversality conditions, calculus of variations, Time scales, Euler--Lagrange equations, infinite horizon..
Mathematics Subject Classification: Primary: 49K05; Secondary: 34N05.
Citation: Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145
References:
[1]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales,", Birkhäuser Boston, (2001).  doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[2]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser Boston, (2003).  doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[3]

M. C. Caputo, Time scales: from nabla calculus to delta calculus and vice versa via duality,, Int. J. Difference Equ., 5 (2010), 25.   Google Scholar

[4]

S. Lang, "Undergraduate Analysis,", 2nd edition, (1997).   Google Scholar

[5]

G. Leitmann, "The Calculus of Variations and Optimal Control,", Mathematical Concepts and Methods in Science and Engineering, (1981).   Google Scholar

[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.  doi: 10.1007/s11590-010-0189-7.  Google Scholar

[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.  doi: 10.3176/proc.2009.4.02.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.015.  Google Scholar

[9]

A. B. Malinowska and D. F. M. Torres, A general backwards calculus of variations via duality,, Optim. Lett., 5 (2011), 587.  doi: 10.1007/s11590-010-0222-x.  Google Scholar

[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.  doi: 10.1016/j.aml.2010.07.013.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.02.022.  Google Scholar

[12]

N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus,, Comput. Math. Appl., 64 (2012), 2166.  doi: 10.1016/j.camwa.2011.12.006.  Google Scholar

[13]

D. F. M. Torres, The variational calculus on time scales,, Int. J. Simul. Multidisci. Des. Optim., 4 (2010), 11.  doi: 10.1051/ijsmdo/2010003.  Google Scholar

show all references

References:
[1]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales,", Birkhäuser Boston, (2001).  doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[2]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser Boston, (2003).  doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[3]

M. C. Caputo, Time scales: from nabla calculus to delta calculus and vice versa via duality,, Int. J. Difference Equ., 5 (2010), 25.   Google Scholar

[4]

S. Lang, "Undergraduate Analysis,", 2nd edition, (1997).   Google Scholar

[5]

G. Leitmann, "The Calculus of Variations and Optimal Control,", Mathematical Concepts and Methods in Science and Engineering, (1981).   Google Scholar

[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.  doi: 10.1007/s11590-010-0189-7.  Google Scholar

[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.  doi: 10.3176/proc.2009.4.02.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.015.  Google Scholar

[9]

A. B. Malinowska and D. F. M. Torres, A general backwards calculus of variations via duality,, Optim. Lett., 5 (2011), 587.  doi: 10.1007/s11590-010-0222-x.  Google Scholar

[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.  doi: 10.1016/j.aml.2010.07.013.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.02.022.  Google Scholar

[12]

N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus,, Comput. Math. Appl., 64 (2012), 2166.  doi: 10.1016/j.camwa.2011.12.006.  Google Scholar

[13]

D. F. M. Torres, The variational calculus on time scales,, Int. J. Simul. Multidisci. Des. Optim., 4 (2010), 11.  doi: 10.1051/ijsmdo/2010003.  Google Scholar

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