## Abstract

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 K_{0}. For a system SGB, a formula is given relating any spectral factor of a (transfer function) coprime fraction power spectral density to K_{0}; a by-product is the description of any normalized coprime fraction of the transfer function in terms of K_{0}. Finally, we describe an alternative way of finding the solution operator K_{0} of the LQ-problem using spectral factorization and a diophantine equation: this is similar to Theorem 2 of Kucera (1981) for lumped systems.

Original language | English |
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Pages (from-to) | 55-75 |

Number of pages | 21 |

Journal | International Journal of Control |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 1990 |