Heat Equation时间最优内部控制的Bang-Bang原理
A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation
摘要：
本文研究了$Ω\times [0,\infty)$中由热方程控制的时间最优内部控制问题。在该问题中，目标集$S$在$L^2(Ω)$中是非空的，控制集$U$在$L^2(Ω)$中是闭的、有界的、非空的，控制函数取自集合$\uad=\{u(\cdot, t): [0,\infty)\ra L^2(Ω) {measurable}; u(\cdot, t)\in U, {a.e. in t} \}$ .我们首先建立了$Ω\times [0,T]$中热方程的零能控性，控制仅限于$Ω$中的一个开非空子集与区间$[0,T]$中的一个正测度子集的乘积集.在此基础上，我们证明了问题的每个最优控制$u^*(\cdot, t)$必然满足bang-bang性质：对于几乎所有的$t\in [0, T^*]$，都满足$u^*(\cdot, t)\in \p U$，其中$\p U$表示集合$U$的边界，$T^*$是最优时间.当目标集$S$是凸的，控制集$U$是闭球时，我们也得到了最优控制的唯一性。
英文摘要：
In this paper, we study a time optimal internal control problem governed by the heat equation in $Ω\times [0,\infty)$. In the problem, the target set $S$ is nonempty in $L^2(Ω)$, the control set $U$ is closed, bounded and nonempty in $L^2(Ω)$ and control functions are taken from the set $\uad=\{u(\cdot, t): [0,\infty)\ra L^2(Ω) {measurable}; u(\cdot, t)\in U, {a.e. in t} \}$. We first establish a certain null controllability for the heat equation in $Ω\times [0,T]$, with controls restricted to a product set of an open nonempty subset in $Ω$ and a subset of positive measure in the interval $[0,T]$. Based on this, we prove that each optimal control $u^*(\cdot, t)$ of the problem satisfies necessarily the bang-bang property: $u^*(\cdot, t)\in \p U$ for almost all $t\in [0, T^*]$, where $\p U$ denotes the boundary of the set $U$ and $T^*$ is the optimal time. We also obtain the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.
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