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 γ 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.