Spectral theory of the linear-quadratic and H problems

Edmond A. Jonckheere, Jyh Ching Juang, Leonard M. Silverman

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12 Citations (Scopus)


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 languageEnglish
Pages (from-to)273-300
Number of pages28
JournalLinear Algebra and Its Applications
Issue numberC
Publication statusPublished - 1989

All Science Journal Classification (ASJC) codes

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


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