Theorem 3.6: bounded continuous control operator

For $y\in H_D^s$ set $v_0=S^{-1}y$ and $N(y)(t)=-i{\mathrm{\Lambda }}_s^{-1}(\phi W(t)v_0)$. Strong continuity of $W$, followed by the bounded cutoff and inverse scale map, shows that $N(y)\in C([0,T];H_D^s)$. Moreover, if $C_{\phi }$ is the cutoff operator norm on $H_D^{-s}$, then $\mathrm{\parallel }N(y){\mathrm{\parallel }}_{C([0,T];H_D^s)}\le \mathrm{\parallel }{\mathrm{\Lambda }}_s^{-1}\mathrm{\parallel }{C}_{\phi }\mathrm{\parallel }{S}^{-1}\mathrm{\parallel }\mathrm{\parallel }y{\mathrm{\parallel }}_{H_D^s}$. Thus $N$ is bounded and complex-linear.

We construct the interior distributional realization used for support. Put $r=\max (0,-s)$. Every test $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$ belongs to $D(A)$: its derivatives through order two are square-integrable weak derivatives by F6, and it belongs to the closure defining $H_0^1$. For $u\in H_D^s$ define $j_{s}u(\eta )={\int }_{\mathrm{\Omega }}u\eta dx$ if $s\ge 0$, and $j_{s}u(\eta )=⟨u,\bar{\eta }{⟩}_{s,-s}$ if $s<0$. The conjugation in the latter formula makes this functional complex-linear in the test. Fix a compact $K\subset \mathrm{\Omega }$ and a bounded box $B$ containing $K$. For tests supported in $K$, the pivot bound and F6 give $\mathrm{\mid }{j}_{s}u(\eta )\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H_D^s}\mathrm{\parallel }\bar{\eta }{\mathrm{\parallel }}_{H_D^r}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H_D^s}\mathrm{\parallel }A\bar{\eta }{\mathrm{\parallel }}_2\le (n+1)\mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{H_D^s}{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 2}\mathrm{\parallel }{\mathrm{\partial }}^{\alpha }\eta {\mathrm{\parallel }}_{\mathrm{\infty }}$. For $s\ge 0$ the first inequality instead follows directly from the $L^2$ inner-product bound and $\mathrm{\parallel }u{\mathrm{\parallel }}_2\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H_D^s}$, with $r=0$. The finite-order criterion makes $j_{s}u$ a distribution, with its test action bounded continuously in $u$. This constructs the needed interior realization without asserting that the map on a negative spectral antidual is injective.

For any $u\in C([0,T];H_D^s)$ and positive integer $m$, let $q_m$ equal $u(kT\mathrm{/}{2}^m)$ on each interval $[kT\mathrm{/}{2}^m,(k+1)T\mathrm{/}{2}^m)$, $0\le k<2^m$, and put $q_m(T)=u(T)$. These are measurable finite-range simple functions. Uniform continuity from F5 gives $\mathrm{\parallel }{q}_m-u{\mathrm{\parallel }}_C\to 0$, so $u$ is strongly measurable. Its norm is measurable as the limit of the simple-function norms, and ${\int }_0^T\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H_D^s}dt\le T\mathrm{\parallel }u{\mathrm{\parallel }}_C<\mathrm{\infty }$. The Bochner criterion therefore gives $u\in L^1(0,T;H_D^s)$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1}\le T\mathrm{\parallel }u{\mathrm{\parallel }}_C$. In particular this applies to every control $N(y)$ in step 1.1.

At nonnegative orders, cutoff multiplication agrees with ordinary multiplication in $L^2$, so $j_s(\phi u)(\eta )=j_{s}u(\phi \eta )$. At negative orders its defining antidual formula gives $⟨\phi u,\bar{\eta }{⟩}_{s,-s}=⟨u,\phi \bar{\eta }{⟩}_{s,-s}=⟨u,\overline{\phi \eta }{⟩}_{s,-s}$, because $\phi$ is real. Thus again $j_s(\phi u)=\phi j_{s}u$ in the distribution convention of F7. If $\eta$ is supported where $\mathrm{\mid }x\mathrm{\mid }<R$, then $\phi \eta =0$, proving spatial vanishing there.

By the definition of $S$, the backward mild solution with control $N(y)$ and terminal value zero has initial value $S{v}_0=y$. Uniqueness of the forward mild solution therefore shows that $N(y)$ drives the initial state $y$ to zero at time $T$.

Put $y=z_0-W(-T)z_T$ and define $G(z_0,z_T)=N(y)$. The group law gives $W(T)y=W(T)z_0-z_T$. By the forward Duhamel formula, the terminal state produced by $G$ from $z_0$ differs from that produced by the same control from $y$ by $W(T)(z_0-y)=z_T$. The latter controlled terminal state is zero by step 3.1, so the former is $z_T$. The group is unitary, and step 1.1 gives $\mathrm{\parallel }G(z_0,z_T){\mathrm{\parallel }}_C\le C(\mathrm{\parallel }{z}_0{\mathrm{\parallel }}_{H_D^s}+\mathrm{\parallel }{z}_T{\mathrm{\parallel }}_{H_D^s})$ for a constant depending on the fixed parameters. The displayed formula follows by substitution.

For $f\in C([0,T];H_D^s)$ define a complex-linear functional on tests $\mathrm{\Psi }\in C_c^{\mathrm{\infty }}(\mathrm{\Omega }\times \mathbb{R};\mathbb{C})$ by $U_f(\mathrm{\Psi })={\int }_0^T{j}_{s}f(t)(\mathrm{\Psi }(\cdot ,t))dt$. The spatial projection $K$ of the test support is compact in $\mathrm{\Omega }$. The slices are continuous in $H_D^2$: all their spatial derivatives through order two vary uniformly continuously on a compact box over any bounded time interval, and the estimate of F6 applied to the difference of two slices bounds its $H_D^2$ norm by those suprema. They are therefore continuous in $H_D^r$. The pairing bound in step 1.2 and continuity of $f$ make the scalar integrand continuous. On each fixed compact space-time support the same estimate gives $\mathrm{\mid }{U}_f(\mathrm{\Psi })\mathrm{\mid }\le T(n+1)\mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_C{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 2}\mathrm{\parallel }{\mathrm{\partial }}_x^{\alpha }\mathrm{\Psi }{\mathrm{\parallel }}_{\mathrm{\infty }}$ for a spatial box $B$ containing its projection. The finite-order criterion, now on $\mathrm{\Omega }\times \mathbb{R}$, makes $U_f$ a distribution. For the control in step 4.1, take $f=\phi h$, which is continuous by cutoff boundedness. Step 2.2 shows $U_f$ vanishes on $\{(x,t):x\in \mathrm{\Omega },\mathrm{\mid }x\mathrm{\mid }<R\}$; its defining time integral also vanishes on $\mathrm{\Omega }\times (\mathbb{R}\setminus [0,T])$. The support definition therefore gives $\mathrm{supp}{U}_f\subseteq \{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }\ge R\}\times [0,T]$. This proves precisely the stated relative space-time support, including the negative-order interpretation.