Turnpike Properties of Pareto Efficient in Cooperative Linear-Quadratic Differential Games

This paper delves into turnpike properties of Pareto efficient in cooperative linear-quadratic differential game (CDG). Firstly, we undertake a transformation of the problem into a fully cooperative differential game (FCDG), predicated on a sufficient condition of Pareto efficient. This transformation illuminates the essence of mutual cooperation inherent in CDG. Subsequently, we derive the optimal equilibrium pair of FCDG by solving a static optimization problem. Additionally, by combining FCDG with traditional optimal control theory, we demonstrate that under the assumption of stabilizability and for a sufficiently large time horizon, the Pareto efficient exhibits turnpike properties. Finally, we conduct an analysis of the time-averaged convergence of the value function in FCDG.