Theorem 1.2: linear control on the full stated scale

PDF p. 4: Theorem 1.2; PDF p. 13: conclusion of Theorem 3.6 proof

Assume the Axiom of Choice and hypothesis (H): $n\ge 2$, the nonempty compact obstacle $\mathrm{\Theta }\subset {\mathbb{R}}^n$ has smooth boundary, its exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ is connected, and its generalized reflected rays leave every compact set in finite time. Let $R>0$ and the real smooth cutoff $\phi$ be as in Control region and nested real cutoffs. Let $T>0$ and $s\in [-2,2]$. For every $z_0,z_T\in H_D^s(\mathrm{\Omega })$, there is a control $h\in C([0,T];H_D^s(\mathrm{\Omega }))$ such that the linear system $i{\mathrm{\partial }}_{t}z+\mathrm{\Delta }z=\phi h$, with homogeneous Dirichlet boundary condition and initial state $z_0$, has a mild solution $z\in C([0,T];H_D^s(\mathrm{\Omega }))$ satisfying $z(T)=z_T$.

Here $H_D^s(\mathrm{\Omega })$ is the Dirichlet spectral Hilbert scale associated with $I-{\mathrm{\Delta }}_D$, and $C([0,T];H_D^s(\mathrm{\Omega }))$ denotes norm-continuous paths. The homogeneous boundary condition is encoded by the Dirichlet realization and its scale; at negative orders the solution is understood in the corresponding mild evolution sense.

The frozen article is cited for attribution of Theorem 1.2. The deduction above uses Theorem 3.6: bounded continuous control operator, rather than the article's statement of Theorem 1.2, as its proof input. The Choice hypothesis is inherited from the stated supplier and is not attributed to the article.