${L}^{\infty}$-null controllability for the heat equation and its consequences for the time optimal control problem.

*(English)* Zbl 1165.93016
Summary: We establish a certain ${L}^{\infty}$-null controllability for the internally controlled heat equation in ${\Omega}\times [0,T]$, with the control 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 obtain a bang-bang principle for the time optimal control of the heat equation with controls taken from the set ${\mathcal{U}}_{\text{ad}}=\{u(\xb7,t):[0,\infty )\to {L}^{2}\left({\Omega}\right)$ measurable; $u(\xb7,t)\in U,$ a.e. in $t\}$, where $U$ is a closed and bounded subset of ${L}^{2}\left({\Omega}\right)$. Namely, each optimal control ${u}^{*}(\xb7,t)$ of the problem satisfies necessarily the bang-bang property: ${u}^{*}(\xb7,t)\in \partial U$ for almost all $t\in [0,{T}^{*}]$, where $\partial U$ denotes the boundary of the set $U$ and ${T}^{*}$ is the optimal time. We also get the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.

##### MSC:

93B05 | Controllability |

93C35 | Multivariable systems, multidimensional control systems |

93C05 | Linear control systems |

35K05 | Heat equation |

49J30 | Optimal solutions belonging to restricted classes (existence) |