Quintic control around nonzero trajectories

PDF p. 21: §5.2(C), first two sentences

Assume the Axiom of Choice. Let $n\ge 2$, $T>0$ and $\lambda \in \mathbb{R}\setminus \{0\}$. Let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact obstacle with smooth boundary and connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, and assume it is non-trapping in the sense of Non-trapping obstacle. Choose the cutoff $\phi$ and control region $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$ as in Linear and nonlinear internally controlled exterior systems, and let $H_D^s(\mathrm{\Omega })$ denote the Dirichlet spectral scale of Dirichlet spectral Sobolev scale and duality.

Let a reference trajectory $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ solve $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{4}w=0$ in the distributional interior sense, with the zero Dirichlet condition supplied by its $H_D^2$ membership. Put $V_1=3\lambda \mathrm{\mid }w{\mathrm{\mid }}^4$ and $V_2=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2{w}^2$. Assume the corresponding unique continuation property: every $q\in C([0,T];H_0^1(\mathrm{\Omega }))$ satisfying $i{\mathrm{\partial }}_{t}q+\mathrm{\Delta }q+V_{1}q-V_2\bar{q}=0$ distributionally in $\mathrm{\Omega }\times (0,T)$ and vanishing on $\omega \times (0,T)$ is identically zero. The boundary condition is understood through $H_0^1$ membership.

Does the following quintic analogue of linearized controllability hold for $s\in \{1,2\}$? For every $z_0,z_T\in H_D^s(\mathrm{\Omega })$, is there $h\in C([0,T];H_D^s(\mathrm{\Omega }))$ and a solution $z\in C([0,T];H_D^s(\mathrm{\Omega }))$ of $i{\mathrm{\partial }}_{t}z+\mathrm{\Delta }z+V_{1}z+V_2\bar{z}=\phi h$, with zero Dirichlet condition and endpoints $z(0)=z_0$, $z(T)=z_T$?

When $n\in \{2,3\}$, does the corresponding nonlinear statement also hold: is there $\delta >0$ such that every $v_0,v_T\in H_D^s(\mathrm{\Omega })$ with $\mathrm{\parallel }{v}_0-w(0){\mathrm{\parallel }}_{H_D^s}\le \delta$ and $\mathrm{\parallel }{v}_T-w(T){\mathrm{\parallel }}_{H_D^s}\le \delta$ can be joined by a solution $v\in C([0,T];H_D^s(\mathrm{\Omega }))$ of $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+\lambda \mathrm{\mid }v{\mathrm{\mid }}^{4}v=\phi h$, for some $h\in C([0,T];H_D^s(\mathrm{\Omega }))$, with zero Dirichlet condition? The equations and endpoint conditions are to be understood in a solution class that makes the distributional equations and boundary and time traces meaningful.

In §5.2(C), the authors suggest that the two quintic analogues should follow along the same lines under the corresponding unique continuation hypothesis; they do not prove those analogues there. Their cited results near a nonzero trajectory concern the cubic equation, while their quintic result is near zero. This item records their proposed extension as an open question, not an established theorem. As of the present authoring pass, no independent worldwide status survey has been performed; the article's formulation and status were checked against its frozen v2 text.

The unique continuation property above is an explicit hypothesis, not a consequence claimed here of non-trapping. The elliptic, time-independent bounded-potential unique continuation result in Weak unique continuation for bounded potentials in three dimensions has different hypotheses and does not by itself supply this time-dependent conjugate-coupled property. No physical premise is used.