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We use geometric invariant theory and the language of quivers to studycompactifications of moduli spaces of linear dynamical systems. A generalapproach to this problem is presented and applied to two well known cases: Weshow how both Lomadze's and Helmke's compactification arises naturally as ageometric invariant theory quotient. Both moduli spaces are proven to be smoothprojective manifolds. Furthermore, a description of Lomadze's compactificationas a Quot scheme is given, whereas Helmke's compactification is shown to be analgebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometricdescription of both compactifications. As an application, we determine thecohomology ring of Helmke's compactification and prove that the twocompactifications are not isomorphic when the number of outputs is positive.

Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems