An Optimal Control Approach to Herglotz Variational Problems

TL;DR

This paper reformulates Herglotz variational problems as optimal control problems, deriving generalized conditions like Euler-Lagrange, transversality, DuBois-Reymond, and Noether's theorem for piecewise smooth functions.

Contribution

It introduces an optimal control framework for Herglotz variational problems, extending classical results to a broader class of functions.

Findings

Derived a generalized Euler-Lagrange equation for Herglotz problems

Established a transversality condition and DuBois-Reymond condition

Proved a version of Noether's theorem for these problems

Abstract

We address the generalized variational problem of Herglotz from an optimal control point of view. Using the theory of optimal control, we derive a generalized Euler-Lagrange equation, a transversality condition, a DuBois-Reymond necessary optimality condition and Noether's theorem for Herglotz's fundamental problem, valid for piecewise smooth functions.