Partial Exponential Turnpike Phenomenon in Linear-Convex Optimal Control

TL;DR

This paper investigates the partial exponential turnpike phenomenon in linear-convex optimal control problems, revealing conditions under which the property holds for certain initial states without requiring controllability or stabilizability.

Contribution

It introduces a refined decomposition of uncontrollable dynamics, provides structural conditions for the partial turnpike property, and characterizes initial states where it applies.

Findings

Partial exponential turnpike property holds for specific initial states.

Explicit characterization of feasible initial states for the turnpike behavior.

Quantification of convergence rate of finite-horizon costs to steady-state value.

Abstract

This paper studies the long-time behavior of optimal solutions for a class of linear-convex optimal control problems. We focus on a partial exponential turnpike property, established without imposing controllability or stabilizability assumptions, where the turnpike behavior holds only for a subset of initial states. By means of a refined decomposition of the completely uncontrollable dynamics, we derive necessary structural conditions for the turnpike property and explicitly characterize the set of feasible initial states. For each such initial state, we associate a static optimization problem whose unique solution determines the corresponding steady state-control pair. For a class of convex stage cost functions, we prove the partial exponential turnpike property and quantify the convergence rate of the averaged finite-horizon optimal cost toward the steady optimal value.