### Abstract

An overview of the spectral theory of the linear-quadratic and H^{∞} problems is presented. Originally, the spectral theory of the linear quadratic problem emerged out of the problem of characterizing the finite-time escape of the Riccati differential equation as a condition on the spectrum of a Toeplitz-plus-Hankel operator. Later and following an independent line of thought, the spectral theory of the H^{∞} problem emerged as a Toeplitz-plus-Hankel operator characterization of the smallest achievable tolerance in feedback design. This common Toeplitz-plus-Hankel operator structure shared by the linear-quadratic and H^{∞} problems is elucidated by mapping the usual H^{∞} frequency-response specification into the time domain, leading to an inequality bound on a quadratic functional precisely induced by the Toeplitz-plus-Hankel operator. With this deep linear-quadratic-H^{∞} connection at hand, we derive a linear-quadratic solution to the H^{∞} problem, and show that such issues as the Adamjan-Arov-Krein problem, pole-zero cancellation in optimal H^{∞} compensation, etc. can be fruitfully attacked using simple linear-quadratic arguments.

Original language | English |
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Pages (from-to) | 273-300 |

Number of pages | 28 |

Journal | Linear Algebra and Its Applications |

Volume | 122-124 |

Issue number | C |

DOIs | |

Publication status | Published - 1989 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

^{∞}problems.

*Linear Algebra and Its Applications*,

*122-124*(C), 273-300. https://doi.org/10.1016/0024-3795(89)90656-3