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Controllability of the Ornstein-Uhlenbeck equation

Abstract:

In this paper we study the controllability of the following controlled Ornstein-Uhlenbeck equation zt = 1/2Δz - 〈x, ∇z〉 + ∑n=1∞ ∑ β =n uβ(t)〈b, hβ〉γd hβ, t > 0, x ∈ ℝd, where hβ is the normalized Hermite polynomial, b ∈ L2(γd), γd(x) = e- x 2/πd/2 is the Gaussian measure in ℝd and the control u ∈ L2(0, t1;l2(γd)), with l2(γd) the Hilbert space of Fourier-Hermite coefficient l2(γd) = { U = {{Uβ} β =n}n≥1: Uβ ∈ ℂ, ∑n=1∞ ∑ β =n Uβ2 < ∞} We prove the following statement: If for all β = (β1, β2,..., βd) ∈ ℕd 〈b, hβ〉γd = ∫Rdbl;d b(x)hβ(x)γd (dx) ≠ = 0, then the system is approximately controllable on [0, t1]. Moreover, the system can never be exactly controllable. Copyright 2006 Oxford University Press.