Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to contain a positive definite matrix (the quadratic hypersurfaces of $A$ and $B$ have trivial intersection).

Has work been done on characterizing the cases when $P(A,B)$ contains a copositive matrix?

UPDATE: Uhlig's paper A recurring theorem about pairs of quadratic forms and extensions: a survey is a convenient reference.

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Can you please give a reference to the Finsler's theorem? –  Pasha Zusmanovich Dec 27 '14 at 10:17
@PashaZusmanovich With pleasure. –  Felix Goldberg Dec 27 '14 at 11:49