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Controllability of stochastic semilinear functional differential equations in Hilbert spaces. (English) Zbl 1038.60056

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

$dX\left(t\right)=\left[-AX\left(t\right)+Bu\left(t\right)+f\left(t,{X}_{t}\right)\right]dt+g\left(t,{X}_{t}\right)dW\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],$

where $W$ is a Hilbert space-valued Wiener process, $A$ generates an analytic semigroup and $f\left(t,·\right)$, $g\left(t,·\right)$ are functions on the path space, ${X}_{t}=\left\{X\left(t+s\right)\left(\omega \right):s\in \left[-r,0\right]\right\}$. 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 $X\left(T,u\right)$, 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.

##### MSC:
 60H15 Stochastic partial differential equations 93E99 Stochastic systems and stochastic control