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

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

2010-07-04

arxiv.org/abs/1007.0584v1

Discrete Contin. Dyn. Syst. 29 (2011), no. 2, 577--593

Mathematics

Optimization and Control

Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-2010

Scientific paper

10.3934/dcds.2011.29.577

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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