IMA Journal of Mathematical Control and Information Advance Access published online on December 14, 2005
IMA Journal of Mathematical Control and Information, doi:10.1093/imamci/dni016
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1 Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101, Venezuela
* To whom correspondence should be addressed. In this paper we study the controllability of the following controlled Ornstein-Uhlenbeck equation
\[
z_t = {1 \over 2}\Delta z - \langle {x, \nabla z}\rangle + \sum\limits_{n = 1}^\infty \sum\limits_{\left| \beta \right| = n} u_{\beta} (t)\langle {b, h_{\beta}}\rangle}_{\gamma_{d}} h_\beta ,\quad t > 0,\ x \in {\mathbb R}^d,
\]
where h
Article
Controllability of the Ornstein-Uhlenbeck equation
Diomedes Bárcenas 1,
Hugo Leiva 1 *,
and
Wilfredo Urbina 2
2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela, Apartado 47195, Los Chaguaramos, Caracas 1041-A, Venezuela
Hugo Leiva, E-mail: Leiva{at}ula.ve
Abstract 
is the normalized Hermite polynomial, b 
L2(
d), d(x) =
\[{e^{-\vert x\vert^2 \over \pi^{d/2}}
\] is the Gaussian measure in \mathbb{R}d and the control u L2(0, t1;l2(d)), with l2(d) the Hilbert space of Fourier-Hermite coefficient
\[
l_2 (\gamma_d) = \left\{U = \{\{U_\beta\}_{\vert \beta \vert = n} \}_{n \ge 1} {:}\ U_\beta \in \mathbb{C}, \sum\limits_{n = 1}^\infty \sum\limits_{\vert \beta \vert = n} \left| U_\beta \right|^2 \le \infty \right\}.
\]
We prove the following statement: If for all = (1, 2, ..., d) \mathbb{N}d
\[
\langle {b,h_\beta}\rangle_{\gamma_d} = \int_{{\mathbb R}^d} {b(x)h_\beta (x)\gamma _d ({\rm d}x) \ne 0,}
\]
then the system is approximately controllable on [0, t1]. Moreover, the system can never be exactly controllable.
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