Partial Exponential Turnpike Phenomenon in Linear-Convex Optimal Control
Jingrui Sun, Lvning Yuan
TL;DR
This paper tackles the long time behavior of linear–convex optimal control problems by proving a partial exponential turnpike phenomenon that holds only for a subset of initial states. It develops a refined Kalman controllable decomposition to identify feasible initial states and links each to an x dependent static problem whose unique solution yields the corresponding steady pair (x^*,u^*). Under strong convexity of the stage cost, both the finite horizon problem (LC)_T and the static problem (O) admit unique solutions and satisfy exponential turnpike estimates for the state and, with extra regularity, for the control, along with a 1/T rate for the averaged cost convergence. The results extend turnpike theory beyond global controllability/stabilizability assumptions and quantify how long time horizons can be effectively approximated by steady solutions that depend on the initial state.
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.