A Deflation Approach for Large-Scale Lur'e Equations

We present an approach to the determination of the stabilizing solution of Lur’e matrix equations. We show that the knowledge of a certain deflating subspace of an even matrix pencil may lead to Lur’e equations which are defined on some subspace, the so-called “projected Lur’e equations.” These projected Lur’e equations are shown to be equivalent to projected Riccati equations, if the deflating subspace contains the subspace corresponding to infinite eigenvalues. This result leads to a novel numerical algorithm that basically consists of two steps. First we determine the deflating subspace corresponding to infinite eigenvalues using an algorithm based on the so-called “neutral Wong sequences,” which requires a moderate number of kernel computations; then we solve the resulting projected Riccati equations. Altogether this method can deliver solutions in low-rank factored form, it is applicable for large-scale Lur’e equations and exploits possible sparsity of the matrix coefficients.