We consider linear control systems in a Hilbert space over an unbounded time interval of the form y'(alpha)(t) = (A - alphaI) y(alpha)(t) + Bu(t), t is an element of (-infinity, T], with bounded control operator B, under appropriate stability assumptions on the operator A. We study how the space of states reachable at time T depends on the parameter alpha greater than or equal to 0. We apply the results to study the dependence on a of the Cameron-Martin spaces of the invariant measures of the Ornstein-Uhlenbeck processes X-alpha defined by the equation driven by the Wiener process W: dX(alpha)(t) = (A - alphaI) X-alpha(t) dt + B dW(t), t greater than or equal to 0.