有限值动态博弈的策略检测及估计问题研究

The optimal control problem of finite-valued dynamical games is studied in "Three Decades of Progress in Control Sciences", Springer, 2010. To solve the optimal control problem, one must solve the strategy detection and estimation problem in advance. Nowadays there are only related results under the assumption of linearity in the literature. In this project, under the framework of the semi-tensor product of matrices, we use a new method based on graph theory and finite automata to solve the general strategy detection and estimation problem. Particularly, we will solve the following problems: (1) determining strategy detectability, (2) input sequence design for detecting the strategy, and (3) strategy estimation. The technical route of this project is: finite-valued dynamical games -> algebraic forms based on the semi-tensor product of matrices -> weighted directed graphs -> finite automata. We hope that in line with this technical route, one will provide effective methods for solving more related problems.

有限值动态博弈的最优控制问题在《控制科学的三十年进展》（Springer，2010）中被研究，而解决该最优控制问题的前提是策略检测及估计问题。现有结果中，只有线性化假设下的策略检测及估计问题被刻画。在本项目中，我们将基于矩阵半张量积，用一种新的图论与有限自动机相结合的方法解决一般性的策略检测及估计问题。具体研究内容如下：(1)策略可检测性问题的判定，(2)检测策略的输入序列设计，(3)策略不可检测情形下的策略估计。本项目的技术路线为：有限值动态博弈->基于矩阵半张量积的代数形式->加权有向图->有限自动机。我们希望沿着这一技术路线，人们能够为更多的相关理论问题提供有效的解决方法。

逻辑动态系统是时间、空间都离散的动态系统,其逆向控制问题(即由输出观测状态的问题,如能观性、可重构性、状态检测等)广泛存在于控制科学及经济学的博弈问题中。在项目完成前尚无一个成熟的方法用来处理这类问题。在本项目中我们建立逆向控制问题与有限自动机的联系,以有限自动机理论为基础建立一套解决逆向控制问题的通用方法。 本项目研究任务的完成将会为未来研究大规模生物学网络的刻画奠定理论基础，从而将本项目中研究结果应用到实际的生物学系统或者人机交互设计研究中。