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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),t[0,T],

where W is a Hilbert space-valued Wiener process, A generates an analytic semigroup and f(t,·), g(t,·) are functions on the path space, X t ={X(t+s)(ω):s[-r,0]}. The basic space is D(A α )= the domain of the fractional power operator A α . 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 α). 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.

60H15Stochastic partial differential equations
93E99Stochastic systems and stochastic control