最优松弛控制问题中的大道性质

Turnpike property ensures theoretically that the long-time optimal control problems can be replaced by the steady-state ones to simplify the analysis and computation, which has been proved in some models from economics, biology, game theory and aerodynamics, etc. However, the study concerning the turnpike property in finite-dimensional nonlinear optimal control problems needs further improvement. Former works are always based on the existence of the optimal control problems, which is not easy to be verified. Also, different works define the turnpike property in different ways, which lacks systematic researches. In this program, finite-dimensional nonlinear optimal relaxed control problems, whose existence can be obtained under weak assumptions, are introduced. As the generalization of the turnpike properties in optimal control problems, three kinds of turnpike properties in optimal relaxed control problems, including integral average turnpike property, mean square turnpike property and exponential turnpike property, are defined. And the program focuses on the sufficient conditions and necessary conditions of these three different turnpike properties, as well as the relation among them. On the basis of this program, new ideas for studying the turnpike properties in optimal control problems are obtained, which helps complete the turnpike theory and simplify the numerical algorithms for solving optimal control problems.

大道性质是利用稳态最优控制问题简化处理定义在长时间区域上的最优控制问题的重要理论依据，在经济学、生物学、博弈论和空气动力学等领域的模型中已得到论证。然而，对于一般的非线性有限维系统的最优控制问题，其大道性质的研究仍有待完善。已有工作通常建立在最优对的存在性上，而存在性本身在许多问题中并不容易验证。同时，不同的工作对大道性质有不同的定义，缺乏较为系统的研究。为此，本项目引入松弛控制，针对存在性容易保证的非线性有限维系统的最优松弛控制问题，定义松弛意义下的三类大道性质：积分平均型、均方型和指数型大道性质，将它们作为经典最优控制问题中的大道性质的推广，分别研究它们成立的充分条件、必要条件以及不同定义之间的关系。通过本项目的研究，为经典最优控制问题中的大道性质提供了新的研究思路，进一步完善了大道性质的理论体系，同时，也为大道性质用于简化数值求解最优控制问题的算法等提供了更丰富的理论基础。

本项目针对最优松弛控制问题中的三类大道性质展开研究，在一定的能控性和凸性等条件下，得到了积分平均型大道性质、均方型大道性质和指数型大道性质成立的充分条件，对简化实际模型中最优控制问题的分析和求解有一定的理论指导意义。同时，本项目考察了以最小成本爆破或熄灭的最优控制问题，论证了其最优解的存在性以及所满足的一阶必要条件。这类问题考察的系统在有限时间内含有奇性，系统无法长时间运行，它让我们得以从另一个角度来加深对长时间区域上最优控制问题中大道性质的理解。本项目在以上两部分内容中均得到了一些有意义的结果，共发表2篇SCI论文。