Noether's symmetry Theorem for variational and optimal control problems with time delay

Gastão Frederico,Delfim Torres

doi:10.3934/naco.2012.2.619

Abstract

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.

Highlights

The concept of symmetry plays an important role in Physics and Mathematics [22]
Typical applications of conservation laws in the calculus of variations and optimal control involve reducing the number of degrees of freedom, reducing the problem to a lower dimension and facilitating the integration of the differential equations given by the necessary optimality conditions [20, 21]
Noether’s theorem asserts that the conservation laws for a system of differential equations that correspond to the Euler–Lagrange equations of a certain variational problem come from the invariance of the variational functional with respect to a parameter continuous group of transformations [24]

Summary

Introduction

The concept of symmetry plays an important role in Physics and Mathematics [22]. Symmetries are described by transformations that applied to a system result in the same object after the transformation is carried out [13]. Noether’s theorem asserts that the conservation laws for a system of differential equations that correspond to the Euler–Lagrange equations of a certain variational problem come from the invariance of the variational functional with respect to a parameter continuous group of transformations [24]. Invariance, symmetries, constants of motion, DuBois– Reymond necessary optimality condition, Noether’s theorem.

Results

Conclusion