Prior interval and rectangle NLS control results

PDF p. 2: §1.2, paragraph beginning For the nonlinear equation

Assume the Axiom of Choice and Countable Choice. The article's introduction reports the following prior work on nonlinear Schrödinger control. Illner, Lange and Teismann [16,17] obtained local exact internal controllability on an interval with periodic boundary conditions in $H_p^1(-\pi ,\pi )$. Lange and Teismann [20] obtained local controllability in $H_0^1$ near a ground state for Dirichlet boundary conditions. Rosier and Zhang [32] are reported there to have extended these results to what the introduction calls “$H^s$ for every $s\ge 0$” and to have treated boundary controls at one endpoint. Laurent [22] obtained semiglobal controllability and stabilization in the one-dimensional setting. For rectangles, Rosier and Zhang [33] combined sharp linear estimates with Bourgain spaces.

The quoted expression “$H^s$ for every $s\ge 0$” reproduces the introduction's notation only. This remark specifies no fractional Sobolev space, domain or boundary convention and invokes no theorem on such a space.

Here $H_p^1(-\pi ,\pi )$ is the $H^1$ space whose endpoint traces agree, and $H_0^1$ is the closure of smooth compactly supported functions in the $H^1$ norm. Exact controllability means that an admissible control steers each allowed initial state to each allowed final state in the specified time; local exact controllability restricts the endpoints to a neighborhood of a reference trajectory.

The terms internal control, boundary control, ground state, semiglobal controllability, stabilization and Bourgain spaces in the historical paragraph are the terminology reported by the introduction; this remark supplies no separate definitions of them. The exact equations, solution and control classes, constraints, time requirements and hypotheses are those of the individual cited works. The present remark records the article's attributions and does not assert a theorem with those omitted hypotheses.

Periodic energy space on an interval

Complex integer-order Sobolev notation

Reachability and observation vocabulary for controlled evolutions

The frozen article is cited for its introductory account and bibliography, not as proof of the underlying prior results. The works it cites have not been independently read for this remark. No equivalence between the different control formulations, nor a specific decay rate, is asserted here.