Noether’s theorem for non-smooth extremals of variational problems with time delay

Abstract

We obtain a non-smooth extension of Noether’s symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler–Lagrange extremals are restricted to those that satisfy the DuBois–Reymond necessary optimality condition. The important case of delayed variational problems with higher order derivatives is considered as well.

Keywords:

• time delays
• invariance
• symmetries
• constants of motion
• conservation laws
• DuBois–Reymond necessary optimality condition
• Noether’s theorem

AMS Subject Classifications:

• 49K05
• 49S05

Acknowledgments

This work was supported by FEDER funds through COMPETE —Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT – Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. Frederico was also supported by FCT through the post-doc fellowship SFRH/BPD/51455/2011, program “Ciência Global”, Odzijewicz and Torres by EU funding under the 7th Framework Programme FP7-PEOPLE-2010-ITN, grant agreement number 264735-SADCO.