Question B: controls on part of the boundary

PDF p. 2: Question B

Assume Countable Choice. Let ${\mathrm{\Omega }}_0\subset {\mathbb{R}}^n$ be a general bounded domain, with $n\ge 1$ and $\mathrm{\Gamma }=\mathrm{\partial }{\mathrm{\Omega }}_0$. Let $\lambda \in \mathbb{R}\setminus \{0\}$, $s\ge 0$ and $T>0$. No obstacle representation of ${\mathrm{\Omega }}_0$ is imposed in Question B.

Question B asks whether the positive answer sought in Question A persists when the boundary control acts only on part of $\mathrm{\Gamma }$: for every pair $u_0,u_1\in H^s({\mathrm{\Omega }}_0)$, can one find such a control $h$ and a solution $u\in C([0,T];H^s({\mathrm{\Omega }}_0))$ with $u(0)=u_0$ and $u(T)=u_1$?

The equation on this general domain is $i{u}_t+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$ on ${\mathrm{\Omega }}_0\times (0,T)$, with either Dirichlet control, prescribing $u=h$, or Neumann control, prescribing ${\mathrm{\partial }}_{\nu }u=h$, on the controlled part of the boundary. Here $i^2=-1$, $u_t=\mathrm{\partial }u\mathrm{/}\mathrm{\partial }t$, $\mathrm{\mid }u{\mathrm{\mid }}^{2}u=u\bar{u}u$, and, for a classically twice differentiable state, $\mathrm{\Delta }u={\sum }_{j=1}^n{\mathrm{\partial }}_{x_j}^{2}u$, acting on real and imaginary parts.

Where the boundary and state admit a classical normal derivative, $\nu$ denotes the outward unit normal and ${\mathrm{\partial }}_{\nu }u={\mathrm{\nabla }}_{x}u\cdot \nu$, componentwise; in particular this convention applies to $C^1$ states up to a $C^1$ boundary. For less regular states an appropriate weak interpretation must be specified separately; no classical or weak boundary trace is asserted for all $s$.

Exact controllability means the stated existence of a steering control for every pair of endpoints; it does not assert a bounded control-selection operator.

For a concrete reading of the state space, use the restriction convention: $H^s({\mathrm{\Omega }}_0)$ consists of restrictions, modulo almost-everywhere equality, of elements of the whole-space Bessel-potential space $H^s({\mathbb{R}}^n)$, with the quotient norm given by the infimum of whole-space norms of extensions. The whole-space space is understood in its weighted Fourier realization. This convention makes the notation definite; it is not an additional controllability result.

The notation $C([0,T];H^s({\mathrm{\Omega }}_0))$ means paths continuous in the $H^s({\mathrm{\Omega }}_0)$ norm, with norm ${\sup }_{0\le t\le T}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^s({\mathrm{\Omega }}_0)}$; the endpoint values in the question are values of such a continuous path.

The question as posed does not specify a particular controlled subset, a boundary condition on its complement, a control space, or a weak trace interpretation for every $s$. A precise instance must specify these, the boundary regularity and the admissible solution class. Those choices can affect the answer; no assertion for every such choice is made here.

Reachability and observation vocabulary for controlled evolutions

Continuous and mixed time-space function spaces

Bessel potentials shift Sobolev order

Restriction Bessel-potential spaces on polyhedral domains

The Axiom of Countable Choice (AC_ω)

Classical normal derivative

The Laplacian of a C² function and of a C² vector field

Exterior and truncated obstacle domains

Canonical higher traces on a bounded boundary or compact interior hypersurface

The Axiom of Choice

The frozen v2 article poses Question B on PDF page 2, before introducing the obstacle geometry as the specialization Question B-prime. Its formulation is retained here as an unresolved question in that article; no independent current-literature status search or claim of worldwide novelty is asserted.

For the partial result reported below, use the truncated geometry ${\mathrm{\Omega }}_0=B_{R_0}(0)\setminus \mathrm{\Theta }$, ${\mathrm{\Gamma }}_1=\mathrm{\partial }{B}_{R_0}(0)$ and ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$, where $\mathrm{\Theta }\subset B_{R_0}(0)$ is nonempty and compact, has a one-sided smooth boundary and connected exterior, and is non-trapping. The smooth-obstacle and non-trapping conventions are those of the geometry supplier: non-trapping means that every complete generalized reflected ray leaves each compact set in finite forward time. These conventions concern this specialized partial result.

For this status paragraph's Sobolev trace convention, assume the Axiom of Choice. For each smooth connected boundary component $S$ contained in ${\mathrm{\Gamma }}_2$ and $s\in \{1,2\}$, use the bounded trace ${\gamma }_S:H^s({\mathrm{\Omega }}_0)\to H^{s-1\mathrm{/}2}(S)$, with the restriction-quotient domain norm and finite-atlas Bessel-potential boundary norm. It agrees with smooth restriction and is independent of whole-space extensions and charts. An endpoint $u_j\in H^s({\mathrm{\Omega }}_0)$ vanishes on ${\mathrm{\Gamma }}_2$ when ${\gamma }_S{u}_j=0$ for every such $S$; this is a condition on its Sobolev equivalence class. The trace supplier applies with $G={\mathrm{\Omega }}_0$, a bounded smooth domain in dimensions two and three, and $s=1$ or $s=2$.

As a reported partial result, the article states in Theorem 1.1 that, for a domain between an enclosing sphere and a smooth nonempty compact non-trapping obstacle with connected exterior, outer-boundary Dirichlet controls steer sufficiently small endpoint data vanishing on the obstacle boundary in every prescribed positive time, in dimensions two and three and for $s=1$ or $s=2$. This sentence attributes the theorem's claim to the article; it does not establish or use that theorem here. The reported result treats the specialized geometry and small data, rather than resolving the general question for arbitrary endpoint data or the Neumann alternative.