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Controllability for blowing up semilinear parabolic equations. (English) Zbl 0952.93061

The paper studies controllability problems (exact and approximate) of a class of semilinear parabolic systems in a bounded domain ${\Omega }\subset {ℝ}^{d}$ described by

${y}_{t}-{\Delta }y+f\left(y\right)=v\left(x,t\right){1}_{\omega },\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }×\left(0,T\right),$
$y=0,\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }×\left(0,T\right),\phantom{\rule{2.em}{0ex}}y\left(x,·\right)={y}_{0}\in {L}^{2}\left({\Omega }\right),\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }·$

Here, $y\left(x,t\right)$ denotes the state; $f:ℝ\to ℝ$ locally Lipschitz continuous; ${1}_{\omega }$ the characteristic function of the nonempty subset $\omega \subset {\Omega }$; and $v\left(x,t\right)\in {L}^{\infty }\left(\omega ×\left(0,T\right)\right)$ the control. Exact controllability of the system is defined as having the following property: Given ${}^{\forall }{y}_{0}$ and ${}^{\forall }{y}^{*}\left(·,t\right)$ (corresponding to ${y}_{0}^{*}$ and ${v}^{*}\left(x,t\right)$), there exists a control $v\left(x,t\right)$ such that $y\left(x,T\right)={y}^{*}\left(x,T\right)$. If ${f}^{\text{'}}\left(s\right)$ is of polynomial growth order and $f\left(s\right)$ is bounded from above by $|s|{log}^{3/2}\left(1+|s|\right)$ at infinity, the exact controllability as well as the approximate controllability are guaranteed. Also the existence of $f$ behaving like $|s|{log}^{p}\left(1+|s|\right)$ at infinity with $p>2$ such that the system fails to be exactly and approximately controllable is shown.

Reviewer: T.Nambu (Kobe)
##### MSC:
 93C20 Control systems governed by PDE 35K20 Second order parabolic equations, initial boundary value problems 93B05 Controllability 35B37 PDE in connection with control problems (MSC2000)