Turnpike Property for Generalized Linear-Quadratic Optimal Control Problem

Abstract

The turnpike phenomenon describes the long time behavior of optimally controlled systems whose optimal trajectories over a sufficiently large time horizon stay for most of the time close to a prescribed trajectory of the system. This thesis is devoted to the characterization of the turnpike property for generalized LQ optimal control problem. Through our research, we derive both sufficient and necessary conditions for the turnpike property in infinite dimensional setting. It is shown that the turnpike property is closely related to certain structural properties of the control system. In particular, we deduce an equivalent condition of the turnpike property in terms of the exponential stabilizability and detectability of the system for finite dimensional case and point spectrum case. We also show in our thesis that the turnpike property for generalized LQ optimal control problem is equivalent to the turnpike property for LQ optimal control problem plus an algebraic condition. Next, we investigate the applications of our results to the generalized LQ optimal control problem subject to the parabolic equations, wave equations, delay equations and in relation with model predictive control schemes.