Real HUM operator for the trajectory linearization

PDF p. 19: Proof of Theorem 1.4, Sw and Gw; PDF p. 20: Gw is defined accordingly

Assume the Axiom of Choice and hypotheses (H) and (W) of Forward and backward real-linear propagators. Thus $T>0$, $\mathrm{\Omega }$ is the smooth connected exterior of the compact nontrapping obstacle, $\lambda \in \mathbb{R}\setminus \{0\}$, and the reference trajectory $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ satisfies the equation specified in (W). Let $\phi$ be the real control cutoff in Linearized equation and its real-duality adjoint. For $s\in \{1,2\}$ put $\sigma =-s$, $E=H_D^s(\mathrm{\Omega })$ and $F=H_D^{\sigma }(\mathrm{\Omega })$. These complex Hilbert spaces are regarded as real Hilbert spaces, with real part of the respective complex inner product. Set ${\mathrm{\Lambda }}_s=(I-{\mathrm{\Delta }}_D{)}^s:E\to F$ as in Scale Riesz map and free HUM operator.

Let $P(t,r):E\to E$ be the homogeneous forward or backward evolution of the first system in Linearized equation and its real-duality adjoint, and let $U(t):F\to F$ be the evolution from time zero of its real-duality adjoint system. For $v_0\in F$, define $v(t)=U(t)v_0$ and $h_{v_0}(t)=-i{\mathrm{\Lambda }}_s^{-1}(\phi v(t))$. Let $z_{v_0}\in C([0,T];E)$ be the unique mild solution of the controlled first system with terminal value $z_{v_0}(T)=0$. Define $S_w:F\to E$ by $S_w{v}_0=z_{v_0}(0)$.

If $S_w$ is a bijection with bounded inverse, define the real-linear exact-control selection $G_w:E\times E\to C([0,T];E)$ by $G_w(x_0,x_T)=h_{v_0}$, where $v_0=S_w^{-1}(x_0-P(0,T)x_T)$. Here $C([0,T];E)$ means norm-continuous paths with the supremum norm. The inverse hypothesis is part of this definition of $G_w$; it is not asserted by the definition of $S_w$.

Forward and backward real-linear propagators

Scale Riesz map and free HUM operator

Linearized equation and its real-duality adjoint

Realification of Hilbert pairings and conjugate-linear coefficients

Continuous and mixed time-space function spaces

The Axiom of Choice

The geometry, trajectory, two equations and propagators are those of Linearized equation and its real-duality adjoint and Forward and backward real-linear propagators. The latter supplies bounded real-linear homogeneous evolutions on $E$ and $F$, including their inverse evolution identities. The full Choice assumption is retained from that evolution result. By Realification of Hilbert pairings and conjugate-linear coefficients, realification gives real Hilbert spaces with the original norms; no complex linearity of the perturbed evolutions is assumed.

By Scale Riesz map and free HUM operator, ${\mathrm{\Lambda }}_s^{-1}:F\to E$ is bounded. The cutoff acts boundedly on $F$ by Cutoff multipliers through order two and by duality, and $t\mapsto U(t)v_0$ is continuous in $F$. Hence $h_{v_0}\in C([0,T];E)$, with a norm bounded by a constant times $\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_F$. The meaning of this continuous-path space is fixed by Continuous and mixed time-space function spaces.

The inhomogeneous mild solution is given by $z(t)=P(t,T)z(T)+{\int }_T^{t}P(t,r)(-i\phi h(r))dr$, with an oriented Bochner integral. The bounded multiplier and the real evolution construction in Bounded real-linear perturbations preserve solution estimates give existence, uniqueness and a uniform bound for continuous $E$-valued controls. In particular, $S_w$ is a well-defined bounded real-linear map. The factor $-i$ remains inside $P(t,r)$; a real-linear evolution need not commute with multiplication by $i$.

Under the additional bounded-inverse hypothesis, let $y=x_0-P(0,T)x_T$ and $v_0=S_w^{-1}y$. The backward solution associated with $h_{v_0}$ starts at $y$ and ends at zero. Adding the homogeneous solution $P(t,0)P(0,T)x_T$ gives a controlled solution starting at $x_0$ and ending at $x_T$, by the evolution identities. Thus the displayed $G_w$ selects a control steering the two endpoints. Its real linearity and boundedness follow by composition of the bounded real-linear maps in its definition. This argument does not establish the bounded-inverse hypothesis.

The original article supplies the constructions and notation on PDF pages 19–20. It is cited for attribution; the well-definedness argument uses the stated mathematical suppliers.