Coercivity and bounded trajectory control operator

First take $a,b$ in the common spectral core and use the control $h_a$ in the backward equation for $z_a$. The data and control are in $H_D^2$: $a\in H_D^2$, multiplication by $\phi$ preserves that space, and ${\mathrm{\Lambda }}_s^{-1}$ increases spectral order. The Duhamel equations, the free group on $D({\mathrm{\Delta }}_D)=H_D^2$, and norm-continuity of the potential operators show that $z_a$ and $U(\cdot )b$ are continuous in $H_D^2$ and continuously differentiable in $L^2$. Indeed the Duhamel integrands are continuous in $H_D^2$; differentiating the free-group convolution in $L^2$ gives the displayed differential equations. Thus the strong pairing identity applies.

Integrating that identity from $0$ to $T$, using $z_a(T)=0$, yields $\mathrm{Re}⟨S_{w}a,b{⟩}_2={\int }_0^T\mathrm{Re}⟨i\phi h_a(t),U(t)b{⟩}_{2}dt$. Since $i{h}_a={\mathrm{\Lambda }}_s^{-1}(\phi U(t)a)$ and $\phi$ is real, the cutoff transpose identity and the scale Riesz identity turn the integrand into $(\phi U(t)a,\phi U(t)b{)}_F$. Both sides are continuous real-bilinear functions of $(a,b)\in F\times F$: use the boundedness of $S_w$, the pivot pairing, the uniform evolution and cutoff bounds, and the finite length of the time interval. Density of the spectral core therefore extends the identity to all $a,b\in F$: $\mathrm{Re}⟨S_{w}a,b{⟩}_{s,-s}={\int }_0^T(\phi U(t)a,\phi U(t)b{)}_{F}dt\mathrm{.}$

Give $Z=L^2(0,T;F)$ the real pairing $(u,v{)}_Z={\int }_0^T(u(t),v(t){)}_{F}dt$. Since $F$ is a real Hilbert space, it is a real Banach space, so the completeness theorem applies to $F$, $p=2$, and the finite interval $(0,T)$ under the assumed Choice. For strongly measurable $u,v$, the real polarization identity $(u(t),v(t){)}_F=(\mathrm{\parallel }u(t)+v(t){\mathrm{\parallel }}_F^2-\mathrm{\parallel }u(t)-v(t){\mathrm{\parallel }}_F^2)\mathrm{/}4$ makes the integrand measurable. Cauchy–Schwarz in $F$ and scalar Hölder show ${\int }_0^T\mathrm{\mid }(u(t),v(t){)}_F\mathrm{\mid }dt\le \mathrm{\parallel }u{\mathrm{\parallel }}_Z\mathrm{\parallel }v{\mathrm{\parallel }}_Z$, so the pairing is finite and independent of representatives. Pointwise bilinearity and symmetry pass to the integral; its diagonal equals $\mathrm{\parallel }u{\mathrm{\parallel }}_Z^2$, which vanishes exactly for the zero class because this is the norm in F6. Thus this pairing is an inner product inducing the complete Bochner norm, and $Z$ is a real Hilbert space.

Define $O:F\to Z$ by $(Oa)(t)=\phi U(t)a$. Each observation orbit is norm-continuous by F3. The finite-range step functions taking its value at the left endpoint of each interval of the partition into $n$ equal pieces converge pointwise to it as $n\to \mathrm{\infty }$, so the orbit is strongly measurable. If $M={\sup }_{0\le t\le T}\mathrm{\parallel }U(t){\mathrm{\parallel }}_{F\to F}$ and $K=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{F\to F}$, then $\mathrm{\parallel }Oa{\mathrm{\parallel }}_Z\le \sqrt{T}KM\mathrm{\parallel }a{\mathrm{\parallel }}_F$. Thus $O$ is bounded and real-linear between the real Hilbert spaces $F$ and $Z$, and observability gives $\mathrm{\parallel }Oa\mathrm{\parallel }\ge C^{-1\mathrm{/}2}\mathrm{\parallel }a{\mathrm{\parallel }}_F$. Let $Q=O^{\ast }O$. The Gramian result makes $Q$ a bounded isomorphism of $F$. By step 1.2 and the scale pairing identity, ${\mathrm{\Lambda }}_s{S}_w=Q$: their real inner products with every $b\in F$ coincide. Since ${\mathrm{\Lambda }}_s:E\to F$ is a bounded isomorphism, $S_w={\mathrm{\Lambda }}_s^{-1}Q$ is a bounded isomorphism from $F$ to $E$. Taking $b=a$ in step 1.2 gives the displayed identity and coercivity.

The inverse just obtained satisfies the additional hypothesis in the definition of $G_w$. For endpoints $x_0,x_T\in E$, that definition takes $a=S_w^{-1}(x_0-P(0,T)x_T)$ and $G_w(x_0,x_T)=h_a$. Its backward controlled solution starts at $x_0-P(0,T)x_T$ and ends at zero; adding the homogeneous solution with terminal value $x_T$ gives initial value $x_0$ and terminal value $x_T$. The boundedness and real linearity of the inverse, propagator and path-valued control map give the corresponding properties of $G_w$. This proves all conclusions.