*(English)*Zbl 0952.93061

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

Here, $y(x,t)$ denotes the state; $f:\mathbb{R}\to \mathbb{R}$ locally Lipschitz continuous; ${1}_{\omega}$ the characteristic function of the nonempty subset $\omega \subset {\Omega}$; and $v(x,t)\in {L}^{\infty}(\omega \times (0,T))$ the control. Exact controllability of the system is defined as having the following property: Given ${}^{\forall}{y}_{0}$ and ${}^{\forall}{y}^{*}(\xb7,t)$ (corresponding to ${y}_{0}^{*}$ and ${v}^{*}(x,t)$), there exists a control $v(x,t)$ such that $y(x,T)={y}^{*}(x,T)$. If ${f}^{\text{'}}\left(s\right)$ is of polynomial growth order and $f\left(s\right)$ is bounded from above by $\left|s\right|{log}^{3/2}(1+|s\left|\right)$ at infinity, the exact controllability as well as the approximate controllability are guaranteed. Also the existence of $f$ behaving like $\left|s\right|{log}^{p}(1+|s\left|\right)$ at infinity with $p>2$ such that the system fails to be exactly and approximately controllable is shown.