本项目拟研究某一类线性抛物系统的精确同步和近似同步控制问题，着重研究以同步为目标的时间最优控制问题。研究内容包括：系统能精确（近似）同步的条件、系统精确（近似）同步时间最优与相应的范数最优控制问题之间的关系、系统精确（近似）同步时间最优控制问题的bang-bang性。相应地，通过本项目的研究，我们期望利用Carleman估计（方程的唯一延拓性）和能控性给出系统能精确（近似）同步的充分条件；在一定条件下，证得精确（近似）同步时间最优控制和范数最优控制问题的等价性；利用前述等价性或某种意义下的时间可测集上的能观性不等式（唯一延拓性）得出精确（近似）同步时间最优控制问题的bang-bang性。

This project aims to study time optimal control and the related problems governed by some kind of linear parabolic systems with exactly synchronized or approximately synchronized constraints. The project focuses on three aspects: the exact(approximate)synchronization of the systems; the relationship between time optimal and norm optimal control problems of the exact(approximate)synchronization of the systems; the bang-bang property of time optimal control problems of the exact(approximate)synchronization of the systems. In this project, we would like to use Carleman estimates(unique continuation)in some sense and the controllability of some linear parabolic systems to obtain the conditions under which the systems are exactly(approximately)synchronized; prove the equivalence between time optimal and norm optimal control problems of the exact (approximate)synchronization of the systems under certain conditions; apply the above mentioned equivalence or the observability inequality (unique continuation) on a measurable set in time to prove the bang-bang property of the time optimal control problems.