Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

Fátima Cruz; Ricardo Almeida; Natália Martins; Fátima Cruz; Ricardo Almeida; Natália Martins

doi:10.3934/math.2021315

AIMS Mathematics

2021, Volume 6, Issue 5: 5351-5369. doi: 10.3934/math.2021315

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Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

Center for Research and Development in Mathematics ad Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

Received: 22 December 2020 Accepted: 10 March 2021 Published: 12 March 2021
MSC : 26A33, 49K05

In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.
- fractional calculus,
- calculus of variations,
- Euler-Lagrange equation,
- isoperimetric problems,
- holonomic problems
Citation: Fátima Cruz, Ricardo Almeida, Natália Martins. Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels[J]. AIMS Mathematics, 2021, 6(5): 5351-5369. doi: 10.3934/math.2021315

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Abstract

In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.

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