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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: $$dX(t)=[-AX(t)+Bu(t)+f(t,X_t)]dt+g(t,X_t)dW(t),\quad t\in[0,T],$$ where $W$ is a Hilbert space-valued Wiener process, $A$ generates an analytic semigroup and $f(t,\cdot)$, $g(t,\cdot)$ are functions on the path space, $X_t=\{X(t+s)(\omega): s\in[-r,0]\}$. The basic space is $D(A^\alpha)=$ 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.

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