On equivalence of pencils from discrete-time and continuous-time control

We study two matrix pencils that arise, respectively, in discrete-time and continuous-time optimal and robust control. We introduce a one-to-one transformation between these two pencils. We show that for the pencils under the transformation, their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related. The eigen-structures of the pencils under consideration have strong connections with the associated control problems. Our result may be applied to connect the discrete-time and continuous-time control problems and eventually lead to a unified treatment of these two types of control problems.