  We develop linear theory for the prediction of excitation wave quenching--the
construction of minimal perturbations which return stable excitations to
quiescence--for localized pulse solutions of models of excitable media. The
theory requires accounting for an additional degree of freedom in the
formulation of the linear theory, and a reconsideration of heuristics for
choosing optimal reference states from their group representation. We compare
the predictions made with the linear theory to direct numerical simulations
across a family of perturbations and assess the accuracy of predictions for
models with distinct stable excitation structures. We find that the theory
achieves qualitative predictive power with only the effort of distinguishing a
root from the asymptotic case, and achieves quantitative predictive power in
many circumstances. Finally, we compare the computational cost of our
prediction technique to other numerical methods for the determination of
transitions in extended excitable systems.
