Necessary conditions for turnpike property for generalized linear-quadratic problems
Roberto Guglielmi, Zhuqing Li
TL;DR
This work addresses the turnpike phenomenon for generalized linear-quadratic problems in infinite-dimensional spaces, connecting long-horizon optimal behavior to core system properties such as exponential stabilizability and exponential detectability. The authors derive several necessary conditions for turnpike behavior, show that the turnpike reference is the unique optimal steady state under mild assumptions, and establish sufficiency in the point-spectrum or finite-dimensional cases. They also prove an equivalence between exponential turnpike properties for generalized LQ and standard LQ problems, including a closed-loop feedback representation guided by a Riccati equation. Collectively, these results extend finite-dimensional turnpike theory to distributed-parameter settings and inform long-horizon design of optimal controls and MPC schemes.
Abstract
In this paper, we develop several necessary conditions of turnpike property for generalizaid linear-quadratic (LQ) optimal control problem in infinite dimensional setting. The term 'generalized' here means that both quadratic and linear terms are considered in the running cost. The turnpike property reflects the fact that over a sufficiently large time horizon, the optimal trajectories and optimal controls stay for most of the time close to a steady state of the system. We show that the turnpike property is strongly connected to certain system theoretical properties of the control system. We provide suitable conditions to characterize the turnpike property in terms of the detectability and stabilizability of the system. Subsequently, we show the equivalence between the exponential turnpike property for generalized LQ and LQ optimal control problems.