Non-smooth unobservable states in control problem for the wave equation in ${\mathbb R}^3$ (corrected)

Abstract

The paper deals with a dynamical system \begin{align*} &u_{tt}-\Delta u=0, \qquad (x,t) \in {\mathbb R}^3 \times (-\infty,0) \\ &u \mid_{|x|<-t} =0 , \qquad t<0\\ &\lim_{s \to \infty} su((s+\tau)\omega,-s)=f(\tau,\omega), \qquad (\tau,\omega) \in [0,\infty)\times S^2\,, \end{align*} where $u=u^f(x,t)$ is a solution ({\it wave}), $f \in {\cal F} :=L_2\left([0,\infty);L_2\left(S^2\right)\right)$ is a {\it control}. For the reachable sets ${\cal U}^\xi:=\{u^f(\cdot, -\xi)\,|\,\, f \in {\cal F}\}\,\,(\xi\geqslant 0)$, the embedding ${\cal U}^\xi \subset {\cal H}^\xi:=\{y \in L_2({\mathbb R}^3)\,|\,\,\,y|_{|x|<\xi}=0\}$ holds, whereas the subspaces ${\cal D}^\xi:={\cal H}^\xi \ominus {\cal U}^\xi$ of unreachable ({\it unobservable}) states are nonzero for $\xi> 0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\cal D}^\xi$ are $C^\infty$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\cal D}^\xi$ with ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=\xi_0>\xi\}$.