Independent generalized reflected-ray and GCC convention

Let $D\subset {\mathbb{R}}^d$, $d\ge 2$, have smooth boundary. A cotangent vector is a linear functional on the tangent space; cotangent vectors over all points form the cotangent bundle. On spacetime $Y={\mathbb{R}}_t\times \bar{D}$, use the wave symbol $p(t,x,\tau ,\xi )=-{\tau }^2+\mathrm{\mid }\xi {\mathrm{\mid }}^2$ and its nonzero characteristic covectors, meaning $p=0$ and $(\tau ,\xi )\mathrm{\ne }0$. In local boundary normal coordinates $y=(y^{\mathrm{\prime }},y_n)$, where $y_n>0$ is the domain side and $y^{\mathrm{\prime }}$ includes time, write $p={\eta }_n^2+r(y,{\eta }^{\mathrm{\prime }})$ and $r_0=r{\mathrm{\mid }}_{y_n=0}$. The compression map is $j(y;\eta )=(y;{\eta }^{\mathrm{\prime }},y_n{\eta }_n)$, with the natural compressed-bundle topology. At the boundary it identifies covectors having the same tangential components, including the incoming and outgoing characteristic covectors. Boundary characteristic points are hyperbolic when $r_0<0$ and glancing when $r_0=0$.

A complete generalized bicharacteristic is a differentiable characteristic curve $\gamma :\mathbb{R}\setminus B\to T^{\ast }Y$, with $B\subset \mathbb{R}$ consisting of isolated reflection parameters, obeying the rules in the External input below. Its compressed image $j\circ \gamma$ has a continuous extension $\widehat{\gamma }$ to all of $\mathbb{R}$. The spatial projection, parametrized by physical time, is a complete generalized ray $x:\mathbb{R}\to \bar{D}$ and travels at unit speed. Curves obeying the same local rules on shorter intervals are ray segments; they are not independently quantified over in GCC or non-trapping. The complete-curve convention does not assert that every locally admissible segment has a complete extension.

For a time-independent open observation region $\omega \subset \bar{D}$ and $T>0$, $(\omega ,T)$ satisfies the geometric control condition (GCC) when every complete generalized ray $x:\mathbb{R}\to \bar{D}$ has $x(t)\in \omega$ for some $t\in (0,T)$. GCC means that such a common $T$ exists. All complete admissible rays are quantified over; uniqueness at infinite-order contact is not asserted.

For a smooth compact obstacle $\mathrm{\Theta }\subset {\mathbb{R}}^d$, its exterior is non-trapping when every complete generalized ray $x:\mathbb{R}\to \overline{{\mathbb{R}}^d\setminus \mathrm{\Theta }}$ leaves every compact set in finite forward time, starting at $t=0$. This is a pointwise condition on complete rays, with no common escape time asserted here.

In the above boundary coordinates put $r_1={\mathrm{\partial }}_{y_n}r{\mathrm{\mid }}_{y_n=0}$ and, for a function $f$ of canonical coordinates, $H_f={\sum }_j({\mathrm{\partial }}_{{\eta }_j}f{\mathrm{\partial }}_{y_j}-{\mathrm{\partial }}_{y_j}f{\mathrm{\partial }}_{{\eta }_j})$. On boundary tangential variables, $H_{r_0}$ uses the same formula with only tangential coordinates and momenta. Put $G^2=G$, the glancing set. For $k\ge 0$, $G^{k+2}$ consists of glancing points where $H_{r_0}^j{r}_1=0$ for $0\le j<k$, and $G^{\mathrm{\infty }}={\bigcap }_{k\ge 0}{G}^{k+2}$. For each integer $\mathrm{\ell }\ge 1$, the diffractive part of $G^{2\mathrm{\ell }}\setminus G^{2\mathrm{\ell }+1}$ is the subset where $H_{r_0}^{2\mathrm{\ell }-2}{r}_1<0$; let $G_d$ be the union of these subsets.

The parameter domain is $\mathbb{R}\setminus B$, where every $s_0\in B$ is isolated in $B$; no reflection parameter is an endpoint of the complete curve. For $s\mathrm{\notin }B$, $\gamma (s)$ is either an interior nonzero characteristic covector or a glancing characteristic covector, and $\gamma$ is differentiable there. It follows $H_p$ in the interior and at $G_d$, and $H_{r_0}$ at $G\setminus G_d$. For each $s_0\in B$ there is $\delta >0$ such that $\gamma (s)$ is interior for $s\in (s_0-\delta ,s_0)\cup (s_0,s_0+\delta )$. The limits ${\rho }^{\pm }={\lim }_{s\to s_0^{\pm }}\gamma (s)$ exist at the boundary, have the same base point and tangential covector, and have opposite nonzero normal covectors. Set $\widehat{\gamma }(s)=j(\gamma (s))$ for $s\mathrm{\notin }B$ and $\widehat{\gamma }(s_0)=j({\rho }^{-})=j({\rho }^{+})$ for $s_0\in B$; the compressed curve $\widehat{\gamma }$ is continuous on $\mathbb{R}$.

The physical-time and spatial coordinates are thus defined also at reflections. The cited construction parametrizes the projected ray by physical time on all of $\mathbb{R}$, at unit speed. These rules include infinite-order glancing points; they do not assert that a point there determines a unique global curve. The boundaryless convention likewise uses complete unit-speed projected geodesics, without reflection.

The non-trapping convention in the Statement is the independent compact-obstacle convention of Burq, Gérard and Tzvetkov. The ray and GCC conventions are the local Euclidean specialization of the definitions of Le Rousseau, Lebeau, Terpolilli and Trélat. No observability theorem, global exterior existence or continuation theorem, or exterior propagation theorem is asserted by these definitions.