The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations
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| Format: | Preprint |
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2025
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| _version_ | 1866918049752285184 |
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| author | Yusubov, Shikhi Sh. Yusubov, Shakir Sh. Mahmudov, Elimhan N. |
| author_facet | Yusubov, Shikhi Sh. Yusubov, Shakir Sh. Mahmudov, Elimhan N. |
| contents | In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< α\leq 1$ and the Riemann- Liouville fractional integral of order $β>0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $α=1$ cannot be adapted to the fractional case $0<α<1$ with final constraints. In spite of this, we prove the Legendre conditions using the standard classical method. The obtained necessary conditions are illustrated by appropriate examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_06736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations Yusubov, Shikhi Sh. Yusubov, Shakir Sh. Mahmudov, Elimhan N. Optimization and Control 26A33, 49K99, 49K05 In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< α\leq 1$ and the Riemann- Liouville fractional integral of order $β>0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $α=1$ cannot be adapted to the fractional case $0<α<1$ with final constraints. In spite of this, we prove the Legendre conditions using the standard classical method. The obtained necessary conditions are illustrated by appropriate examples. |
| title | The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations |
| topic | Optimization and Control 26A33, 49K99, 49K05 |
| url | https://arxiv.org/abs/2506.06736 |