Question B prime: outer boundary control

PDF p. 2: Question B′

Assume countable choice. Let $n\ge 2$ and let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact obstacle with smooth boundary, connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, and the non-trapping property of Non-trapping obstacle: every complete admissible generalized reflected ray $x:\mathbb{R}\to \bar{\mathrm{\Omega }}$, parametrized by physical time at unit speed, leaves each compact set in finite forward time. Choose $R_0>0$ with $\mathrm{\Theta }\subset B_{R_0}(0)$, and set ${\mathrm{\Omega }}_0=B_{R_0}(0)\setminus \mathrm{\Theta }$, ${\mathrm{\Gamma }}_1=\mathrm{\partial }{B}_{R_0}(0)$ and ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$.

For given $s\ge 0$, $T>0$ and arbitrary $u_0,u_1\in H^s({\mathrm{\Omega }}_0)$, can one find a control $h$ on ${\mathrm{\Gamma }}_1\times (0,T)$ and a solution $u\in C([0,T];H^s({\mathrm{\Omega }}_0))$ with $u(0)=u_0$ and $u(T)=u_1$ to the cubic equation $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$, where $\lambda \in \mathbb{R}\setminus \{0\}$, satisfying either the Dirichlet conditions $u=h$ on ${\mathrm{\Gamma }}_1$ and $u=0$ on ${\mathrm{\Gamma }}_2$, or the Neumann conditions ${\mathrm{\partial }}_{\nu }u=h$ on ${\mathrm{\Gamma }}_1$ and ${\mathrm{\partial }}_{\nu }u=0$ on ${\mathrm{\Gamma }}_2$?

Here $H^s({\mathrm{\Omega }}_0)$ is the restriction space of $H^s({\mathbb{R}}^n)$, equipped with the infimum extension norm. The path space consists of norm-continuous functions of time. The normal $\nu$ is outward from ${\mathrm{\Omega }}_0$. The question leaves the control class and any weak interpretation of the boundary conditions to a formulation in which they are meaningful; it does not assert trace existence for arbitrary states.

Only the function-space formulation is justified here. By Real-order Bessel-potential completion Hˢ, under Countable Choice the whole-space completion $H^s({\mathbb{R}}^n)$ is complex Banach. For $s\ge 0$, Restriction Bessel-potential spaces on polyhedral domains supplies its linear injection ${\iota }_s$ into $L^2({\mathbb{R}}^n)$ with $\mathrm{\parallel }{\iota }_{s}F{\mathrm{\parallel }}_2\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^s}$. Set $RF=({\iota }_{s}F){\mathrm{\mid }}_{{\mathrm{\Omega }}_0}$. Restrictions and vector operations are well-defined on almost-everywhere classes by Complex Holder, Minkowski, and the quotient norm. Monotonicity of the nonnegative integral in Monotonicity and nonnegative homogeneity of the nonnegative integral gives $\mathrm{\parallel }RF{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_0)}^2\le \mathrm{\parallel }{\iota }_{s}F{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}^2\le \mathrm{\parallel }F{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)}^2\mathrm{.}$ Consequently $R$ is bounded and linear. Its kernel $M$ is closed: if $F_j\in M$ and $F_j\to F$ in $H^s$, then $\mathrm{\parallel }RF{\mathrm{\parallel }}_2=\mathrm{\parallel }R(F-F_j){\mathrm{\parallel }}_2\le \mathrm{\parallel }F-F_j{\mathrm{\parallel }}_{H^s}\to 0$, so $RF=0$. Two extensions have the same restriction precisely when their difference lies in $M$. The displayed infimum-extension norm in the question is therefore the quotient norm of $H^s({\mathbb{R}}^n)\mathrm{/}M$. Under the stated Countable Choice, A quotient of a Banach space by a closed subspace is Banach makes it a definite complete norm. This verifies the Banach-target hypothesis of Continuous and mixed time-space function spaces for the path space used in the question. No boundary condition is built into this restriction quotient, and this argument makes no controllability assertion.

Here the ray quantifier uses complete curves defined for all real times, as in the non-trapping supplier; locally admissible ray segments are not separately quantified over, and no theorem extending every segment to a complete ray is assumed.

The frozen v2 article poses this question with arbitrary endpoints. It reports a positive local Dirichlet result for $n\in \{2,3\}$ and $s\in \{1,2\}$, for sufficiently small endpoint data satisfying the obstacle-boundary compatibility condition, and for every $T>0$; it discusses the Neumann problem separately. This is a restricted result, not a claimed answer to the full question above. The question is recorded as unresolved in the article; no independent current-status review is claimed here.

Exact controllability means that every pair of endpoints in the specified admissible set can be joined by an admissible control and solution. The article's question uses all of $H^s({\mathrm{\Omega }}_0)$ as its endpoint set, whereas any rigorous answer must specify the admissible control and solution classes and the meanings of the boundary conditions.