Manifold Turnpikes, Trims and Symmetries

TL;DR

This paper extends the classical turnpike theory to manifold settings motivated by mechanical systems with symmetries, providing new conditions for existence and illustrating with the Kepler problem.

Contribution

It introduces the concept of manifold turnpikes, generalizes optimality conditions to symmetry-induced manifolds, and proposes dissipativity-based criteria for their existence.

Findings

Necessary optimality conditions on manifolds match reduced problems.

Sufficient conditions for manifold turnpikes are established.

The Legendre transformation is extended to adjoint variables.

Abstract

Classical turnpikes correspond to optimal steady states which are attractors of optimal control problems. In this paper, motivated by mechanical systems with symmetries, we generalize this concept to manifold turnpikes. Specifically, the necessary optimality conditions on a symmetry-induced manifold coincide with those of a reduced-order problem under certain conditions. We also propose sufficient conditions for the existence of manifold turnpikes based on a tailored notion of dissipativity with respect to manifolds. We show how the classical Legendre transformation between Euler-Lagrange and Hamilton formalisms can be extended to the adjoint variables. Finally, we draw upon the Kepler problem to illustrate our findings.