A structure-preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear-quadratic infinite time horizon optimal control. We focus on small-scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete-time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure-preserving doubling algorithm. Implementation issues such as the choice of the parameter gamma in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill-posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright (C) 2015 John Wiley & Sons, Ltd.