Non static exponential turnpike property for optimal control problems with symmetries and boundary conditions

TL;DR

This paper proves an exponential convergence property for optimal control solutions with symmetries, using a reduced problem approach and hyperbolicity assumptions, with applications to orbital transfer and other problems.

Contribution

It establishes an exponential trim turnpike property for control problems with Abelian Lie group symmetries via a novel reduced Hamiltonian framework.

Findings

Optimal trajectories stay exponentially close to trim primitives near boundaries.

The approach applies to linear, nonlinear, and orbital transfer problems.

Hyperbolicity of the reduced Hamiltonian equilibrium is key to the results.

Abstract

Optimal control problems with symmetries often admit a non stationary turnpike property called trim turnpike, which characterizes the convergence of optimal solutions to certain symmetry induced trajectories called trim primitives. In this paper we establish an exponential trim turnpike property for a class of optimal control problems with structural properties related to Abelian Lie group symmetries. The key ingredient of our approach is the introduction of an appropriate reduced optimal control problem. We show that extremals of the original problem can be characterized through a reduced Hamiltonian boundary value problem that coincides with the optimality system of the reduced problem. Under a hyperbolicity assumption on the equilibrium of the corresponding reduced Hamiltonian system we prove that optimal trajectories remain exponentially close, up to boundary layers near the endpoints, to a trim primitive defined by the static reduced problem. The theoretical results are illustrated on three representative examples: linear and nonlinear problems with quadratic cost and the Kepler orbital transfer problem.