Controllability of the Ornstein-Uhlenbeck equation

Abstract:

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

Año de publicación:

2006

Keywords:

• Approximate controllability
• Ornstein-Uhlenbeck equation
• Compact semigroup

Fuente:

scopus