Let \(H\) be a separable Hilbert space, \({\mathfrak P}(H)\) the space of probability measures defined on the Borel field of subsets of \(H\), and \(C^k(H)\) the space of \(k\) times Fréchet differentiable functions on \(H\). The author studies the control problem associated with a measure-valued function \(\mu_t\) satisfying
\[ (d/dt)\langle\mu_t, \pi\rangle= \langle\mu_t, L^u(\mu_t)\varphi\rangle,\quad t\in [0,T], \]
\[ \mu_0= v\in{\mathfrak P}(H), \]
where \(f\in C(H)\) is a smooth test function, \(u: I\times H\to E\) a control law, \(E\) a separable, Banach space, and \(\{L^u(\mu); \mu\in{\mathfrak P}\}\) a family of second-order differential operators on \(C^2(H)\),
\[ L^u(\mu)\varphi(x)= (1/2)\text{Tr}(QD^2\varphi)+ (A^* D\varphi, x)+ (f(x, \mu), D\varphi)+ (g(x, u), D\varphi), \]
where \(A: D(A)\to H\), \(D(A)\subset H\), is the infinitesimal generator of a \(C_0\)-semigroup of bounded linear operators on \(H\), and \(f: H\times{\mathfrak P}(H)\to H\), \(g: H\times E\to H\) are suitable maps. The author shows that under certain conditions the control problem has a unique solution \(\{\mu^u_t: t\in [0, T]\}\) for each given initial \(v\in{\mathfrak P}(H)\) and each control \(u\). Several typical cases and some physically motivated examples are considered.