Isoperimetric problems of the calculus of variations with fractional derivatives

Almeida, Ricardo; Ferreira, Rui A. C.; Torres, Delfim F. M.

doi:10.1016/s0252-9602(12)60043-5

Cited by 22 publications

(18 citation statements)

References 43 publications

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“…Due to this assumption, our optimal control problem can not be replaced with a fractional problem of calculus of variations and studied with the aid of the methods of the fractional calculus of variations presented, for example, in monograph (cf. also ). We obtain our main result using the smooth–convex extremum principle due to Ioffe and Tikchomirov.…”

Section: Resultsmentioning

confidence: 91%

Pontryagin maximum principle for fractional ordinary optimal control problems

Kamocki

2013

Math Methods in App Sciences

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Section: Resultsmentioning

confidence: 91%

Pontryagin maximum principle for fractional ordinary optimal control problems

Kamocki

2013

Math Methods in App Sciences

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“…Next, it can be proven that F satisfies the fractional EulerLagrange equation and that in case the extremizer does not satisfies the Euler-Lagrange associated to , then we can take λ 0 = 1 (cf. [14]). In conclusion, the first variation of F evaluated along an extremal must vanish, and so we obtain a system similar to (6), replacing L by F .…”

Section: A Numerical Methods To the Fractional Calculus Of Variationsmentioning

confidence: 99%

A discrete time method to the first variation of fractional order variational functionals

2013

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Abstract:The fact that the first variation of a variational functional must vanish along an extremizer is the base of most effective solution schemes to solve problems of the calculus of variations. We generalize the method to variational problems involving fractional order derivatives. First order splines are used as variations, for which fractional derivatives are known. The Grünwald-Letnikov definition of fractional derivative is used, because of its intrinsic discrete nature that leads to straightforward approximations. PACS

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“…Theorem 8. Let the pair (x * , T * ) be local minimum for J as in (7). If there exist and are continuous the functions t → D α,ψ…”

Section: Variational Principles With Delaymentioning

confidence: 99%

Optimality conditions for fractional variational problems with free terminal time

Almeida¹

2018

Discrete &Amp; Continuous Dynamical Systems - S

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This paper provides necessary and sufficient conditions of optimality for a variational problem involving a fractional derivative with respect to another function. Fractional Euler-Lagrange equations are proven for the fundamental problem and when in presence of an integral constraint. A Legendre condition, which is a second-order necessary condition, is also obtained. Other cases, such as the infinite horizon problem, with delays in the Lagrangian, and with high-order derivatives, are considered. A necessary condition that the optimal fractional order must satisfy is proved.Mathematics Subject Classification 2010: 26A33, 49K05, 34A08.

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Isoperimetric problems of the calculus of variations with fractional derivatives

Cited by 22 publications

References 43 publications

Pontryagin maximum principle for fractional ordinary optimal control problems

Pontryagin maximum principle for fractional ordinary optimal control problems

A discrete time method to the first variation of fractional order variational functionals

Optimality conditions for fractional variational problems with free terminal time

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