Linear Control Systems on Unbounded Time Intervals and Invariant Measures of Ornstein--Uhlenbeck Processes in Hilbert Spaces

We consider linear control systems in a Hilbert space over an unbounded time interval of the form y′ α(t) = (A− αI)yα(t) +Bu(t), t ∈ (−∞, 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 α ≥ 0. We apply the results to study the dependence on α of the Cameron–Martin spaces of the invariant measures of the Ornstein–Uhlenbeck processes Xα defined by the equation driven by the Wiener process W : dXα(t) = (A− αI)Xα(t) dt+B dW (t), t ≥ 0.