Controllability of the Ornstein-Uhlenbeck equation

In this paper we study the controllability of the following controlled Ornstein–Uhlenbeck equation z t = 1 2 z − −x, ∇z + ∞ n=1 |β|=n u β (t)b, h β γ d h β , t > 0, x ∈ R d , where h β is the normalized Hermite polynomial, b ∈ L 2 (γ d), γ d (x) = e −|x| 2 π d/2 is the Gaussian measure in R d and the control u ∈ L 2 (0, t 1 ; l 2 (γ d)), with l 2 (γ d) the Hilbert space of Fourier–Hermite coefficient l 2 (γ d) =