A note on "Families of linear-quadratic problems : continuity properties"

In the paper "Families of Linear-Ouadratic Problems: Continuity properties" one-parameter (e) families of these problems with stability are investigated. In this technical note we will show that the set of invariant zeros of the e-problem is contained in the set of invariant zeros of the "boundary" problem (the problem with e=0) if the "boundary" system is left invertible. This result is obtained only by applying the assumptions on the continuity and the monotonicy of the weighting matrices that are made in the above-mentioned paper. Consequently, we can replace the third assumption in that paper by a much weaker one.