该项目拟开展对于一类分布参数系统的最优控制问题，研究其最优解在大时间尺度下的渐进行为。主要目标是希望导出这类问题的最优解满足指数型周期turnpike性。值得指出的是研究最优控制的长时间渐进行为不仅具有数学理论意义，而且还有广泛的应用前景。特别地，在运用打靶法求解最优控制的数值计算过程中，它起到很大的帮助作用。我们希望从Pontryagin最大值原理出发，通过运用算子Riccati方程和Lyapunov方程的周期解，构造一个线性同胚变换去研究最优解的稳定性，并最终得到最优解的指数周期turnpike性。

This project is devoted to study the long time asymptotic behavior of optimal controls for distributed parameter systems under some suitable conditions. The main purpose of this project is to establish the periodic and exponential turnpike property for solutions of these optimal control problems. Studying the long time behavior of optimal controls is important from the perspective of both mathematical theory and applications. This property is particularly useful in order to implement successfully a numerical shooting method for solving the optimality system. To obtain the periodic and exponential turnpike property, we hope to design a periodic and linear homeomorphism transformation, based on periodic solutions of the Riccati and Lyapunov operator equations, to analyze the periodic stability of optimality system resulting from the Pontryagin maximum principle.