Entry and leaving arcs of turnpikes: their exact computation in the calculus of variations

TL;DR

This paper provides a precise method for computing entry and leaving arcs of turnpikes in one-dimensional autonomous variational problems using phase space analysis, with an approximation algorithm and example.

Contribution

It introduces an exact computation technique for turnpike arcs in variational calculus, leveraging phase space and transversality conditions, with an accompanying approximation algorithm.

Findings

Exact computation of turnpike arcs achieved

An approximation algorithm is developed

Illustrative example demonstrates the method

Abstract

We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated to the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that that the transversality condition is (generically) a curve. An approximation algorithm is presented and an example included for completeness.