On Computing Stable Lagrangian Subspaces of Hamiltonian Matrices and Symplectic Pencils∗

This paper presents algorithms for computing stable Lagrangian invariant subspaces of a Hamiltonian matrix and a symplectic pencil, respectively, having purely imaginary and unimodular eigenvalues. The problems often arise in solving continuousor discrete-time H∞-optimal control, linear-quadratic control and filtering theory, etc. The main approach of our algorithms is to determine an isotropic Jordan subbasis corresponding to purely imaginary (unimodular) eigenvalues by using the associated Jordan basis of the square of the Hamiltonian matrix (the S +S−1-transformation of the symplectic pencil). The algorithms preserve structures and are numerically efficient and reliable in that they employ only orthogonal transformations in the continuous case.