Endpoint feedback around a trajectory

PDF p. 20: definitions of Nw, hz and Tw

Assume the Axiom of Choice. Let $n\ge 2$, $T>0$, and let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty smooth compact nontrapping obstacle. This is hypothesis (H): every generalized reflected ray in the exterior leaves every compact set in finite time, with the generalized reflected-ray convention of the exterior geometry. Choose $R>0$ with $\mathrm{\Theta }\subset B_R(0)$ and a real-valued $\phi \in C^{\mathrm{\infty }}({\mathbb{R}}^n;[0,1])$ such that $\phi (x)=0$ for $\mathrm{\mid }x\mathrm{\mid }\le R$, $\phi (x)>0$ for $\mathrm{\mid }x\mathrm{\mid }>R$, and $\phi (x)=1$ for $\mathrm{\mid }x\mathrm{\mid }\ge R+1$. Put $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$. Let $\lambda \in \mathbb{R}\setminus \{0\}$ and let $w$ satisfy (W): $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ and $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ distributionally. Assume (UCPw): every $C([0,T];H_0^1(\mathrm{\Omega }))$ solution of the adjoint equation $i{v}_t+\mathrm{\Delta }v+2\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}v-\lambda w^2\bar{v}=0$ that vanishes on $(0,T)\times \omega$ is identically zero.

Take $s\in \{1,2\}$ and set $E=H_D^s(\mathrm{\Omega })=D((I-{\mathrm{\Delta }}_D{)}^{s\mathrm{/}2})$, with its spectral norm. Here ${\mathrm{\Delta }}_D$ is the Dirichlet Laplacian. Regard $E$ as a real Banach space. Let $S(t,r)$ be the real-linear homogeneous evolution for $i{z}_t+{\mathrm{\Delta }}_{D}z+2\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}z+\lambda w^2\bar{z}=0$. Let $G_w:E\times E\to C([0,T];E)$ be the bounded real-linear selector which steers this linearized system between its two input endpoints with control $h$ appearing as $\phi h$. Use the same $R$, $\omega$ and $\phi$ for (UCPw), $G_w$ and the control multiplier.

For $z\in C([0,T];E)$ such that $F(w,z)=\lambda (2w\mathrm{\mid }z{\mathrm{\mid }}^2+\bar{w}{z}^2+\mathrm{\mid }z{\mathrm{\mid }}^{2}z)$ is strongly measurable and belongs to $L^1(0,T;E)$, and for prescribed endpoints $z_0,z_T\in E$, define

$N_w(z)(t)={\int }_0^{t}S(t,r)(iF(w,z)(r))dr,h_z=G_w(z_0,z_T-N_w(z)(T)),$

and

${\mathcal{T}}_w(z)(t)=S(t,0)z_0+N_w(z)(t)+{\int }_0^{t}S(t,r)(-i\phi h_z(r))dr\mathrm{.}$

The $L^1(0,T;E)$ space uses Lebesgue time measure and almost-everywhere classes of strongly measurable maps with integrable $E$ norm. The integrals are Bochner integrals in the realification of $E$, and $C([0,T];E)$ denotes norm-continuous paths. Thus the definition gives $N_w(z),{\mathcal{T}}_w(z)\in C([0,T];E)$ and $h_z\in C([0,T];E)$. It applies to any chosen control space $X_T\subset C([0,T];E)$ once its members satisfy the displayed remainder condition; invariance of $X_T$ and contraction are separate assertions.

Exterior and truncated obstacle domains

Control region and nested real cutoffs

Continuous and mixed time-space function spaces

Reference trajectory hypothesis W

Conditional energy-level unique continuation hypothesis

Dirichlet spectral Sobolev scale and duality

Quadratic-cubic remainder after subtracting a trajectory

Duhamel formula for a real-linear propagator

Coercivity and bounded trajectory control operator

Cutoff multipliers through order two and by duality

The Axiom of Choice

The exterior geometry, one-sided smooth-obstacle and generalized reflected-ray conventions, and nontrapping hypothesis (H) are those of Exterior and truncated obstacle domains. The radius, observation region and smooth real cutoff are chosen as in Control region and nested real cutoffs, consistently for (UCPw), the selector and the multiplier. The trajectory hypothesis is that of Reference trajectory hypothesis W, and the displayed unique-continuation implication is the conditional hypothesis of Conditional energy-level unique continuation hypothesis. The spectral scale, Dirichlet operator and complete normed state space are supplied by Dirichlet spectral Sobolev scale and duality, under the Choice assumption recorded in The Axiom of Choice. The expression for $F$ and the meaning of the trajectory remainder are supplied by Quadratic-cubic remainder after subtracting a trajectory.

The strong-measurability convention, Lebesgue-time $L^1(0,T;E)$ space and Bochner integration are supplied by Continuous and mixed time-space function spaces; its countable-choice assumption is covered by the stated Axiom of Choice.

Under these hypotheses, Coercivity and bounded trajectory control operator supplies the bounded real-linear selector $G_w$ and its endpoint-steering property. The evolution and the continuous path obtained by integrating an $L^1(0,T;E)$ forcing are supplied by Duhamel formula for a real-linear propagator. Hence the assumed remainder condition makes $N_w(z)$ a continuous $E$-valued path, so $N_w(z)(T)$ is a valid input to $G_w$. The resulting $h_z$ is continuous in $E$. By Cutoff multipliers through order two and by duality, multiplication by $\phi$ is bounded on $E$; consequently $-i\phi h_z$ is continuous and therefore integrable on the finite interval. The same evolution result makes the last integral and ${\mathcal{T}}_w(z)$ continuous. The factors $i$ and $-i$ remain inside $S(t,r)$ because this evolution is only assumed real-linear.

The original article supplies the notation and definitions on PDF page 20. It is cited for attribution; the stated suppliers justify the domain and well-definedness. No nonlinear existence or contraction claim is made by this definition.