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A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation | | Gengsheng Wang
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9 Dec 2006 | | Subject: | Optimization and Control | | Abstract: | In this paper, we study a time optimal internal control problem governed by the heat equation in $Omega imes [0,infty)$. In the problem, the target set $S$ is nonempty in $L^2(Omega)$, the control set $U$ is closed, bounded and nonempty in $L^2(Omega)$ and control functions are taken from the set $uad={u(cdot, t): [0,infty)
a L^2(Omega) {measurable}; u(cdot, t)in U, {a.e. in t} }$. We first establish a certain null controllability for the heat equation in $Omega imes [0,T]$, with controls restricted to a product set of an open nonempty subset in $Omega$ 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 $tin [0, T^*]$, 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. | | Source: | arXiv, math/0612237 | | Services: | Forum | Review | PDF | Favorites | |
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