Solutions of generalized discrete-time Lyapunov equations

This paper investigates a constrained generalized discrete-time Lyapunov equation (GDLE). It is shown that the constrained GDLE admits a (hermitian) solution if and only if the underlying pencil is admissible and some augmented simplectic pencil has the complementarity property. Explicit formulae, expressed in terms of the eigenstructures of the underlying pencil and the augmented simplectic pencil, for all solutions of the constrained GDLE are derived. A numerical example is also given for illustration.