该项目拟开展关于时变线性抛物方程的时间最优控制问题的研究。主要是希望导出时间最优控制的bang-bang性，次之，是希望利用bang-bang性得到最优时间与最优控制的更深刻的刻画，后者可能通过建立不同最优控制问题的等价性得到。Bang-bang性是时间最优控制的一重要性质，对它的研究是有理论意义和应用背景的。时不变的线性抛物方程的bang-bang性可由时间可测集上的能观性不等式得到。然而，一般的时变线性抛物方程的时间最优控制的bang-bang性不能由上述不等式得到。因此，我们希望能发现新的方法，导出这一性质。

In this project, we will study time optimal controls for time-varying linear parabolic equations. The main purpose is to build up the bang-bang property for time optimal controls. Besides we hope to provide more delicate characterization for the optimal time and the optimal control through utilizing the bang-bang property. The latter might be obtained provided that we could establish some equivalence for different kinds of optimal control problems. The bang-bang property is an important property for time optimal controls. Studies on this property is significant from perspective of both mathematical theory and applications. When systems are time invariant linear parabolic equations, the bang-bang can be derived from the observability estimate from measurable sets in time. However, this inequality cannot imply the bang-bang property for the case when systems are time-varying linear parabolic equations in general. Theorfore, we wish to find a new way to build up this property for the current case.