Partial Exponential Turnpike Phenomenon in Linear-Convex Optimal Control

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.