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Controllability of the Ornstein-Uhlenbeck equation. (English) Zbl 1106.93007

Summary: We study the controllability of the following controlled Ornstein-Uhlenbeck equation \[ z_t=\frac 12\Delta z-\langle x,\nabla z\rangle+ \sum^\infty_{n=1} \sum_{|\beta|=n}u_\beta(t)\langle b,h_\beta \rangle_{\gamma_d}h_\beta,\quad t>0,\;x\in\mathbb{R}^d, \] where \(h_\beta\) is the normalized Hermite polynomial, \(b\in L^2 (\gamma_d)\), \(\gamma_d(x)=\frac{e^{-|x|^2}}{\pi^{d/2}}\) is the Gaussian measure in \(\mathbb{R}^d\) and the control \(u\in L^2(0,t_1;l_2 (\gamma_d))\), with \(l_2 (\gamma_d)\) the Hilbert space of Fourier-Hermite coefficient \[ l_2(\gamma_d)= \left\{U=\bigl\{\{U_\beta\}_{| \beta|=n}\bigr\}_{n\geq 1}:U_\beta\in\mathbb{C}, \sum^\infty_{n=1} \sum_{|\beta|=n}|U_\beta|^2<\infty\right\}. \] We prove the following statement: If for all \(\beta=(\beta_1,\beta_2,\dots,\beta_d)\in \mathbb{N}^d\) \[ \langle b,h_\beta\rangle_{\gamma_d}=\int_{\mathbb{R}^d} b(x)h_\beta(x) \gamma_d(dx)\neq 0, \] then the system is approximately controllable on \([0,t_1]\). Moreover, the system can never be exactly controllable.

MSC:

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
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