Let $M$ be a smooth manifold (without boundary) and $S \subseteq M$ be a smooth embedded submanifold with non-empty boundary $\partial S$. Does there exist a smooth, embedded extension $\tilde S$ of $S$ in the sense that $\tilde S$ is an embedded, smooth submanifold of $M$ without boundary of same dimension as $S$ and $S \subset \tilde S$?
I am quite certain that there is one, but I have not been able to prove it. I also have not found anything on google and don't have any advanced books on differential topology lying around. The proposition would be of use to prove smoothness of mappings defined on $S$, so maybe there's been done something like that before. Either way, those were the two approaches I tried:
1) Use the existence of boundary slice charts and canonically extend the slices of $S$ in each such chart. As long as the extension is "small enough" there should be no issue regarding non-smooth intersections of the different slices and mutual "bad touching" preventing embeddedness. My idea was to use the tubular neighborhood theorem to find a sufficiently "nice" neighborhood of $\partial S$ preventing those bad effects, but I gave up here.
2) Use the flowout theorem with a (smooth) vector field tangent to $S$ and outward pointing on $\partial S$ to "generate" $\tilde S$. My problem here is that the flowout theorem as I know it is too weak, i.e. it does not yield a smooth embedding.