Lur’e equations and even matrix pencils

In this work, we consider the so-called Lur’e matrix equations that arise e.g. in model reduction and linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils.