Optimality conditions for the calculus of variations with higher-order delta derivatives

Ferreira, Rui A. C.; Malinowska, Agnieszka B.; Torres, Delfim F. M.

doi:10.1016/j.aml.2010.08.023

Cited by 22 publications

(16 citation statements)

References 33 publications

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“…The calculus of variations on time scales is an important subject under strong current research (see [10,13,19,35,36,41] and references therein). Here we review a general necessary optimality condition for problems of the calculus of variations on time scales [37,38].…”

Section: Resultsmentioning

confidence: 99%

The variational calculus on time scales

Torrest

2010

Int. J. Simul. Multidisci. Des. Optim.

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Abstract. The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.

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Section: Resultsmentioning

confidence: 99%

The variational calculus on time scales

Torrest

2010

Int. J. Simul. Multidisci. Des. Optim.

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“…In Section 2.1, we give basic recalls on time scale calculus. Section 2.2 is devoted to recalls on nonshifted calculus of variations on general time scales developed in [12,13,19]. In particular, the ∆-integral Euler-Lagrange equation (EL ∆ int ) is given as a necessary condition for local optimizers of nonshifted Lagrangian functionals, see Proposition 3.…”

Section: Nonshifted Calculus Of Variations On Time Scales With ∇Diffementioning

confidence: 99%

“…In this section, we recall some results on nonshifted calculus of variations on general time scales provided in [12,13,19]. Let L be a Lagrangian i.e.…”

Section: Recalls On Nonshifted Calculus Of Variations On General Timementioning

confidence: 99%

Nonshifted calculus of variations on time scales with ∇-differentiable σ

Bourdin

2014

Journal of Mathematical Analysis and Applications

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In calculus of variations on general time scales, an Euler-Lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted Lagrangian functionals, see e.g. [R.A.C. Ferreira and co-authors, Optimality conditions for the calculus of variations with higher-order delta derivatives, Appl. Math. Lett., 2011]. In this paper, we prove that the ∇-differentiability of the forward jump operator σ is a sharp assumption on the time scale in order to ∇-differentiate this integral Euler-Lagrange equation. This procedure leads to an Euler-Lagrange equation of differential form. Furthermore, from this differential form, we prove a Noether-type theorem providing an explicit constant of motion for Euler-Lagrange equations admitting a symmetry.

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“…There exist several different approaches to deal with nondifferentiability in problems of the calculus of variations: the time scale approach, which typically deals with delta or nabla differentiable functions [1][2][3][4][5][6][7][8][9][10]; the fractional approach, which deals with fractional derivatives of order less than one [11][12][13][14][15][16][17][18][19][20]; the quantum approach, which deals with quantum derivatives [21][22][23][24][25][26]; and the scale derivative approach recently introduced by J. Cresson in 2005 [27].…”

Section: Introductionmentioning

confidence: 99%