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Interior controllability of the $nD$ semilinear heat equation. (English) Zbl 1243.93020

Summary: In this paper, we prove the interior approximate controllability of the following semilinear heat equation

$\left\{\begin{array}{cc}{z}_{t}\left(t,x\right)={\Delta }z\left(t,x\right)+{1}_{\omega }u\left(t,x\right)+f\left(t,z,u\left(t,x\right)\right)\hfill & \text{in}\phantom{\rule{1.em}{0ex}}\left(0,\tau \right]×{\Omega },\hfill \\ z=0,\hfill & \phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{1.em}{0ex}}\left(0,\tau \right)×\partial {\Omega },\hfill \\ z\left(0,x\right)={z}_{0}\left(x\right),\hfill & x\in {\Omega },\hfill \end{array}\right\$

where ${\Omega }$ is a bounded domain in ${ℝ}^{N}\left(N\ge 1\right),{z}_{0}\in {L}^{2}\left({\Omega }\right)$, $\omega$ is an open nonempty subset of ${\Omega }$, and ${1}_{\omega }$ denotes the characteristic function of the set $\omega$. The distributed control $u$ belong to $\in {L}^{2}\left(\left[0,\tau \right];{L}^{2}\left({\Omega };\right)\right)$ and the nonlinear function $f:\left[0,\tau \right]×ℝ×ℝ\to ℝ$ is smooth enough and there are $a,b,c\in ℝ$, with $c\ne -1$, such that

$\underset{\left(t,z,u\right)\in {Q}_{\tau }}{sup}|f\left(t,z,u\right)-az-cu-b|<\infty ,$

where ${Q}_{\tau }=\left[0,\tau \right]×ℝ×ℝ$. Under this condition, we prove the following statement: For all open nonempty subset $\omega$ of ${\Omega }$ the system is approximately controllable on $\left[0,\tau \right]$. Moreover, we could exhibit a sequence of controls steering the nonlinear system (1) from an initial state ${z}_{0}$ to an $ϵ$ neighborhood of the final state ${z}_{1}$ at time $\tau >0$, which is very important from a practical and numerical point of view.

##### MSC:
 93B05 Controllability 93C25 Control systems in abstract spaces 35K05 Heat equation