*(English)*Zbl 1038.60056

The authors investigate the following semilinear controlled equation with delay in a Hilbert space:

where $W$ is a Hilbert space-valued Wiener process, $A$ generates an analytic semigroup and $f(t,\xb7)$, $g(t,\xb7)$ are functions on the path space, ${X}_{t}=\{X(t+s)\left(\omega \right):s\in [-r,0]\}$. The basic space is $D\left({A}^{\alpha}\right)=$ the domain of the fractional power operator ${A}^{\alpha}$. It is proved that under Lipschitz and growth conditions on $f$, $g$, approximate controllability of the deterministic system implies approximate controllability of the stochastic one (i.e., the ${L}_{p}$-closure of possible values of $X(T,u)$, as $u$ varies, is the whole ${L}_{p}$, $p$ being related to $\alpha $). A criterion of exact controllability is also given (the assumption on the compactness of the semigroup is then dropped). Finally, an application to a stochastic heat equation is shown.