Spectral Factorization by Symmetric Extraction for Distributed Parameter Systems

Abstract

The spectral factorization problem of a scalar coercive spectral density is considered in the framework of the Callier--Desoer algebra of distributed parameter system transfer functions. Criteria are given for the infinite product representation of a meromorphic coercive spectral density of finite order and for the convergence of infinite product representations of spectral factors, i.e., for the convergence of the symmetric extraction method for solving the spectral factorization problem of such spectral density. These convergence criteria are applied to the solution of the linear-quadratic optimal control problem by spectral factorization for a specific class of semigroup Hilbert state-space systems with a Riesz-spectral generator. The speed of convergence of the symmetric extraction method is also considered. As an example a damped vibrating string model is handled.