Linear quadratic regulator and spectral factorization for continous-time descriptor systems

The authors study the linear quadratic optimal regulator problem and a related spectral factorization for continous-time descriptor systems. They first derive a generalized Riccati differential equation for the finite-horizon problem based on the two-point boundary value problem for the Hamiltonian equation. They then obtain an optimal feedback control and the optimal cost without performing a preliminary transformation for a descriptor system. For the infinite-horizon problem, it is shown that one can easily compute a solution of the steady-state generalized algebraic Riccati equation (GARE) and an optimal feedback gain by using the generalized eigenvectors and the generalized principal vectors of the generalized eigenvalue problem. Furthermore, applying GARE, the authors discuss a spectral factorization algorithm for a positive definite matrix described by a descriptor form. The factorization results suggest the possibility of a spectral factorization for transfer matrices which are not proper