Spectral factorization and LQ-optimal regulation for multivariable distributed systems

A necessary and sufficient condition is proved for the existence of a bistable spectral factor (with entries in the distributed proper-stable transfer function algebra A in the context of distributed multi variable convolution systems with no delays; a by-product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system (with entries in the algebra of fractions over A We next study semigroup state-space systems SGB with bounded sensing and control (having a transfer function with entries in £V) and consider its standard LQ-optimal regulation problem having an optimal state feedback operator K₀. For a system SGB, a formula is given relating any spectral factor of a (transfer function) coprime fraction power spectral density to K₀; a by-product is the description of any normalized coprime fraction of the transfer function in terms of K₀. Finally, we describe an alternative way of finding the solution operator K₀ of the LQ-problem using spectral factorization and a diophantine equation: this is similar to Theorem 2 of Kucera (1981) for lumped systems.