本项目拟开展关于Fitzhugh-Nagumo方程和抛物型方程组的时间最优控制的研究，其中，控制作用在空间区域的内部，控制约束为逐点约束。逐点约束不仅在理论上是时间最优控制中的一类重要约束情形，而且还是实际应用包括数值计算中常常使用的一种情形。本项目的目的是通过建立Fitzhugh-Nagumo方程时间最优控制的最大值原理，导出时间最优控制和最优时间满足的充分必要条件，从而为时间最优控制和最优时间的数值计算实现奠定坚实的理论基础，并尝试对其进行数值计算。另一方面，希望通过对抛物型方程组时间最优控制的存在性和最大值原理的研究，了解在方程组中施加一个控制与两个控制在时间最优控制问题上的差异，得到相应的两个最优时间差的定量刻画，这样，不仅填补了时间最优控制问题在这一方面上的空白，而且为时间最优控制在实际中的应用提供了理论上的支持。毕竟，施加多的控制意味着大量的人力、物力和财力的投入。

This project is concerned with time optimal controls of Fitzhugh-Nagumo equation and parabolic systems, where the controls are acted on interior of the domain. Moreover, the control constraint is in the pointwise form. Pointwise constraint is not only an important constraint of time optimal control problems, but also a frequently used constraint in applications including numerical calculations. By establishing maximum prinicple of time optimal controls of Fitzhugh-Nagumo equation, we aim to derive necessary and sufficient conditions of time optimal controls and optimal time. Therefore, solid theoretical basis can be laid for numerical calculation of time optimal controls and optimal time. Then we attempt to calculate them. Moreover, by studying existence of time optimal controls and maximum principle of time optimal control problems governed by parabolic systems, we hope to understand differences between time optimal control problems when one or two controls are acted on, and then obtain the relations between two optimal times. Thus, we not only fill the blank of time optimal control problems on this issue, but also provide theoretical support for time optimal controls in practical applications. After all, exerting more controls means enormous investment of human, material and financial resources.