A non-standard class of variational problems of Herglotz type

Natália Martins

doi:10.3934/dcdss.2021152

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2022, Volume 15, Issue 3: 573-586. Doi: 10.3934/dcdss.2021152

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A non-standard class of variational problems of Herglotz type

Natália Martins^,

Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

^* Corresponding author: natalia@ua.pt
^* Corresponding author: natalia@ua.pt

Received: January 31, 2020

Revised: January 31, 2021

Early access: November 2021

Published: March 2022

This work is supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020

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Abstract

In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function $ x $ and its derivative and an unknown functional $ z $, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.

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Mathematics Subject Classification: Primary: 49K15, 49K05, 34H05; Secondary: 93C23.

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