Endpoint-correcting nonlinear feedback map

PDF p. 14: Theorem 3.7 proof, definitions of N, hv and T

Assume the Axiom of Choice and hypothesis (H): $n\ge 2$, $\mathrm{\Theta }\subset {\mathbb{R}}^n$ is nonempty and compact with smooth boundary, its exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ is connected, and its generalized reflected rays leave every compact set in finite time. Let $T>0$, $s\in [1,2]$, $m\in \{3,5\}$, and let $\phi$ be the real control cutoff associated with this exterior. Put $H=H_D^s(\mathrm{\Omega })$, $W(t)=e^{it{\mathrm{\Delta }}_D}$, and $g(z)=\lambda \mathrm{\mid }z{\mathrm{\mid }}^{m-1}z$ for real nonzero $\lambda$. Let $X_T$ satisfy the abstract perturbation hypotheses: it is a real or complex Banach space continuously included in $C([0,T];H)$ with inclusion norm at most one, and its homogeneous evolution, retarded forcing operator and nonlinearity satisfy conditions (i)–(iii) of that definition with a constant $C_0\ge 1$.

Let $G:H\times H\to C([0,T];H)$ be the bounded linear endpoint-to-control operator: for each pair $(a,b)$, the mild solution of the linear controlled equation with initial state $a$ and control $G(a,b)$ has terminal state $b$. For prescribed $v_0,v_T\in H$ and $v\in X_T$, define

$N(v)(t)={\int }_0^{t}W(t-r)(ig(v)(r))dr,h_v=G(v_0,v_T-N(v)(T)),$

and define the feedback map $\mathcal{T}:X_T\to X_T$ by

$\mathcal{T}(v)(t)=W(t)v_0+N(v)(t)+{\int }_0^{t}W(t-r)(-i\phi h_v(r))dr\mathrm{.}$

All integrals are Bochner integrals in $H$. Equivalently, the constant factors $i$ and $-i$ may be written outside the respective integrals as $H$-valued equalities. The definition itself asserts neither a fixed point nor uniqueness of a controlled nonlinear solution.

Exterior and truncated obstacle domains

Control region and nested real cutoffs

Dirichlet spectral Sobolev scale and duality

Abstract nonlinear control-space hypotheses

Theorem 3.6: bounded continuous control operator

Cutoff multipliers through order two and by duality

Continuous and mixed time-space function spaces

The Axiom of Choice

Average convergence for a continuous Banach-valued function

Linearity of the Bochner integral

The geometry and non-trapping convention are those of Exterior and truncated obstacle domains, and the cutoff is that of Control region and nested real cutoffs. The Dirichlet realization, spectral Hilbert space $H=H_D^s(\mathrm{\Omega })$ and unitary group $W(t)=e^{it{\mathrm{\Delta }}_D}$ are those of Dirichlet spectral Sobolev scale and duality. The Banach path space, retarded operator and nonlinear estimates are specified by Abstract nonlinear control-space hypotheses; its continuous inclusion makes evaluation at $T$ a bounded map from $X_T$ to $H$. The time-space and Bochner-integral conventions are those of Continuous and mixed time-space function spaces. Choice is retained as required by the stated hypotheses and the selector theorem The Axiom of Choice. In particular, the assumed Axiom of Choice supplies a choice function for every countable family of nonempty sets, as required by the continuous-path integrability lemma used below.

By the nonlinear hypothesis, $g(v)\in L^1(0,T;H)$. Multiplication by $i$ is a bounded scalar operation on the complex Hilbert space $H$, so $ig(v)$ belongs to the same $L^1$ space. The integrability of this scalar multiple and homogeneity of the integral follow from Linearity of the Bochner integral in the complex Banach space $H$; complex linearity of $W(t-r)$ then justifies the equivalent placement of scalar factors in the Definition. The retarded forcing hypothesis therefore gives $N(v)\in X_T$ and $N(v)(0)=0$; the latter follows because the Bochner integral over the zero-length interval is zero. Evaluation at $T$ makes the second argument of $G$ an element of $H$. Under the stated geometry, time, scale and Choice assumptions, Theorem 3.6: bounded continuous control operator supplies the bounded selector $G$ and its continuous $H$-valued control $h_v$.

The cutoff multiplication operator is bounded on $H$ by Cutoff multipliers through order two and by duality. Thus $r\mapsto -i\phi h_v(r)$ is continuous as an $H$-valued path and belongs to $L^1(0,T;H)$ on the finite interval. Here continuous-path Bochner integrability is supplied by Average convergence for a continuous Banach-valued function, applied with $X=H$, $a=0$ and $b=T$. The retarded forcing hypothesis places its integral in $X_T$, and the homogeneous hypothesis places $W(\cdot )v_0$ in $X_T$. Their sum with $N(v)$ is consequently an element of $X_T$. This construction does not require multiplication by $i$ to preserve $X_T$ when $X_T$ is regarded only as a real space.

The frozen article, PDF page 14, is cited for attribution of the definitions of $N$, $h_v$ and $\mathcal{T}$. Their well-definedness above uses the stated mathematical suppliers.