Controllability of stochastic semilinear functional differential equations in Hilbert spaces.
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.
semilinear stochastic functional differential equation
scientific article; zbMATH DE number 2056372