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}-Δu=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+τ)ω,-s)=f(τ,ω), \qquad (τ,ω) \in [0,\infty)\times S^2\,, \end{align*} where $u=u^f(x,t)$ is a solution ({\it wave}), $f\in \mathcal{F}:=L_2\left([0,\infty );L_2\left(S^2\right)\right)$ is a {\it control}. For the reachable sets ${\mathcal{U}}^{\xi }:=\{u^f(\cdot ,-\xi )|f\in \mathcal{F}\}(\xi ⩾0)$, the embedding ${\mathcal{U}}^{\xi }\subset {\mathcal{H}}^{\xi }:=\{y\in L_2({\mathbb{R}}^3)|y{|}_{|x|<\xi }=0\}$ holds, whereas the subspaces ${\mathcal{D}}^{\xi }:={\mathcal{H}}^{\xi }\ominus {\mathcal{U}}^{\xi }$ of unreachable ({\it unobservable}) states are nonzero for $\xi >0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\mathcal{D}}^{\xi }$ are $C^{\infty }$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\mathcal{D}}^{\xi }$ with $\mathrm{sing}\mathrm{supp}h\subset \{x\in {\mathbb{R}}^3||x|={\xi }_0>\xi \}$.

arXiv comment2 figures