Conservation laws for invariant functionals containing compositions§

§Presented orally at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), in Pretoria, South Africa, 22–24 August, 2007.

Abstract

The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler–Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work, we prove a generalization of the necessary optimality condition of DuBois–Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic.

§Presented orally at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), in Pretoria, South Africa, 22–24 August, 2007.

Keywords:

• Variational calculus
• Functionals containing compositions
• Symmetries
• DuBois–Reymond condition
• Noether's theorem

Keywords:

• AMS Subject Classifications: 49K05
• 49J05

Acknowledgement

This work is part of first author's PhD project, supported by the Portuguese Institute for Development (IPAD). Second author was supported by the Centre for Research on Optimization and Control (CEOC) through the Portuguese Foundation for Science and Technology (FCT), cofinanced by the European Community fund FEDER/POCTI. The authors are grateful to Paweł Góra who shared Chapter 4 of 2.

Notes

§Presented orally at the 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), in Pretoria, South Africa, 22–24 August, 2007.