Euler-Lagrange equationsfor composition functionals in calculus of variations on timescales

Agnieszka B. Malinowska; Delfim F. M. Torres

doi:10.3934/dcds.2011.29.577

Article Contents

2011, Volume 29, Issue 2: 577-593. Doi: 10.3934/dcds.2011.29.577

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Euler-Lagrange equations for composition functionals in calculus of variations on time scales

Agnieszka B. Malinowska^1, and
Delfim F. M. Torres^2,

1.
Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok
2.
Department of Mathematics, University of Aveiro, 3810-193 Aveiro

Received: April 30, 2009

Revised: February 28, 2010

Published: May 2011

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Abstract

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H (\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.

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Mathematics Subject Classification: Primary: 49K05, 39A12; Secondary: 49K99.

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