Trim turnpikes for optimal control problems with symmetries

TL;DR

This paper extends the exponential turnpike property to trim turnpikes in optimal control problems with symmetries, using geometric reduction techniques and illustrating with classical mechanical systems.

Contribution

It introduces the concept of exponential trim turnpikes for control systems with symmetries, generalizing previous static and manifold turnpike results.

Findings

Established the exponential trim turnpike theorem for symmetric control systems.

Applied the theory to Kepler and Rigid body problems.

Demonstrated the effectiveness of geometric reduction in analyzing symmetric control problems.

Abstract

Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing symmetries. Following recent works, which generalized the classical concept of static turnpike to manifold turnpike, we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the Rigid body problem.