Necessary endpoint compatibility for boundary controls

PDF p. 4: Remark 2

Assume the Axiom of Choice. In the truncated smooth obstacle geometry, let $n\in \{2,3\}$, $s\in \{1,2\}$ and $T>0$. Suppose $u\in C([0,T];H^s({\mathrm{\Omega }}_0))$ has endpoints $u(0)=u_0$ and $u(T)=u_1$, and $h\in C([0,T];H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_1))$. If the outer Dirichlet control conditions ${\gamma }_{{\mathrm{\Gamma }}_1}u(t)=h(t)$ and ${\gamma }_{{\mathrm{\Gamma }}_2}u(t)=0$ hold for every $0<t<T$, then necessarily ${\gamma }_{{\mathrm{\Gamma }}_2}{u}_0={\gamma }_{{\mathrm{\Gamma }}_2}{u}_1=0$, $h(0)={\gamma }_{{\mathrm{\Gamma }}_1}{u}_0$ and $h(T)={\gamma }_{{\mathrm{\Gamma }}_1}{u}_1$. These are equalities in the corresponding boundary Sobolev spaces.

The article states that the control constructed in its proof, obtained as the outer trace of the extended solution, satisfies these identities. Its Theorem 1.1 permits every $T>0$ and does not require $\mathrm{\Theta }$ to be star-shaped.

The original article is cited for the attribution in the second paragraph of the Statement. The proof above is a conditional consequence of bounded traces and norm continuity; it neither establishes the article's control construction nor uses the article's assertion as a premise. No star-shapedness or non-trapping assumption is needed for this conditional endpoint deduction.