A Stochastic Fractional Calculus with Applications to Variational Principles

Zine, Houssine; Torres, Delfim F. M.

doi:10.3390/fractalfract4030038

Cited by 13 publications

(10 citation statements)

References 22 publications

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“…Here special cases are presented in which global optimality of solutions of variational problems are guaranteed when satisfying the Euler-Lagrange necessary conditions. The emphasis here is on this sense of the word 'global' in contrast to global descriptions of geometry or dynamics which can be found elsewhere [6,11,12].…”

Section: Nonstandard Case Studies In Global Optimalitymentioning

confidence: 99%

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Bootstrapping Globally Optimal Variational Calculus Solutions

Chirikjian¹

2021

Preprint

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Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems, uniqueness and global optimality are generally hard to prove, and often times they may not even exist. This is also true for variational problems on Lie groups, with the Euler-Poincaré equation establishing necessary conditions. This article therefore reviews several cases where unique globally optimal solutions can be guaranteed, and establishes a "bootstrapping" process to build new globally optimal variational solutions on larger spaces from existing ones on smaller spaces. This general theory is then applied to several topics such as optimal framing of curves in three-dimensional Euclidean space, optimal motion interpolation, and optimal reparametrization of video sequences to compare salient actions.

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Section: Nonstandard Case Studies In Global Optimalitymentioning

confidence: 99%

“…Moreover, from ( 14), J1 is globally minimized on the same global minimizer of J2, namely q * (t) and the minimal values are related as (12).…”

Section: J2(q)mentioning

confidence: 99%

Bootstrapping Globally Optimal Variational Calculus Solutions

Chirikjian¹

2021

Preprint

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“…We embark on this section by briefly introducing the essential structure of fractional calculus and fractional operators, as well as some important inequalities of stochastic calculus. For the more salient details on these matters, see the textbooks [36,37,38,39,40] and research paper [41,42]. We introduce the classical fractional operators.…”

Section: Fractional and Stochastic Calculusmentioning

confidence: 99%

Some Results on Backward Stochastic Differential Equations of Fractional Order

Mahmudov¹,

Ahmadova²

2021

Preprint

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Our aim in this paper is to deal with a new type differential equation so-called Caputo fractional backward stochastic differential equations (for short Caputo fBSDEs) and study the global existence and uniqueness of an adapted solution to Caputo fBSDEs of order α ∈ ( 1 2 , 1) whose coefficients satisfy Lipschitz condition by applying fundamental lemma which plays a crucial role in the theory of Caputo fBSDEs. The interesting point here is to use a new weighted norm in square-integrable measurable function space for fundamental lemma and therefore for the whole paper. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and mild solution.

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“…In 2019, Abdeljawad et al [15] developed a fractional integration by parts formula for Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. In 2020, Zine and Torres [16] introduced a stochastic fractional calculus, and obtained a stochastic fractional Euler-Lagrange equation. Motivated by these works, particularly [14][15][16][17], and with the help of our weighted generalized fundamental integration by parts formula, we extend the available Euler-Lagrange equations.…”

Section: Introductionmentioning

confidence: 99%

Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation

Zine

Lotfi

Torres

et al. 2022

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Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.

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A Stochastic Fractional Calculus with Applications to Variational Principles

Cited by 13 publications

References 22 publications

Bootstrapping Globally Optimal Variational Calculus Solutions

Bootstrapping Globally Optimal Variational Calculus Solutions

Some Results on Backward Stochastic Differential Equations of Fractional Order

Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation

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