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Given the Lyapunov matrix equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A'Q + QA = -P</tex> a fundamental inequality which is satisfied by the extremal eigenvalues of the matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> , provided <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> is a stability matrix, is established. This result, besides being interesting from a theoretical standpoint, is extremely useful in the determination of suboptimal controllers for the minimum time problem [1].
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E.Y. Shapiro (1974) studied this question.
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