Non-trapping obstacle

PDF p. 2: footnote 1; PDF p. 3: Remark 1

Let $n\ge 2$, and let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact obstacle with smooth boundary and connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$. A complete generalized ray is the spatial projection $x:\mathbb{R}\to \bar{\mathrm{\Omega }}$, parametrized by physical time at unit speed, of a complete admissible generalized bicharacteristic in the exterior, including reflected and glancing boundary behavior.

The obstacle is non-trapping if every such complete ray leaves every compact subset of ${\mathbb{R}}^n$ in finite forward time: for each ray $x(t)$ and each compact set $K\subset {\mathbb{R}}^n$, there is a finite $t\ge 0$ with $x(t)\mathrm{\notin }K$. The time may depend on both the ray and $K$. This is the non-trapping part of the article's standing hypothesis (H).

Exterior and truncated obstacle domains

Independent generalized reflected-ray and GCC convention

The geometry and the meanings of compactness, smooth boundary and connected exterior are fixed by Exterior and truncated obstacle domains. The generalized-ray convention is fixed independently by Independent generalized reflected-ray and GCC convention: it includes complete compressed reflected and glancing curves and identifies their spatial projections. The quantified curves have time domain $\mathbb{R}$; the escape test uses their forward restrictions. A curve defined only on a finite interval or on $[0,\mathrm{\infty })$ is not automatically a complete ray in this predicate: applying the predicate to that curve requires a complete extension. The condition above is therefore a definite predicate on the obstacle and those admissible rays. It asks for a finite escape time separately for each ray and compact set; this definition does not assert a common escape time, uniqueness at infinite-order contact, or a geometric control theorem.

The article's footnote on page 2 describes reflected geometric-optics rays leaving compact sets in finite time, and Remark 1 uses generalized bicharacteristics. The article is cited for attribution; the ray convention used here is supplied by the independent prerequisite.