A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation
| dc.creator | Wang, Gengsheng | |
| dc.date | 2006-12-09 | |
| dc.date.accessioned | 2026-07-07T07:34:47Z | |
| dc.date.available | 2026-07-07T07:34:47Z | |
| dc.description | 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. | |
| dc.description | 22 pages | |
| dc.identifier | https://arxiv.org/abs/math/0612237 | |
| dc.identifier | http://arxiv.org/abs/math/0612237 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/119890 | |
| dc.subject | Optimization and Control | |
| dc.subject | 93C35; 93C05 | |
| dc.title | A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation | |
| dc.type | text |