From J.J. Sylvester to Adolf Hurwitz: A Historical Review
In this paper, a brief historical review of the works of four illustrious mathematicians in the nineteenth century — namely, Hermite (1854), Routh (1877), Lyapunov (1892), and Hurwitz (1895) — which affected research on stability in the twentieth century is mentioned.
Based on the work of Sylvester (1840), the connection between the Hurwitz criterion and resultants is established. This leads to a derivation of the generalized and specialized Orlando’s formulae. The connection between Hurwitz and Routh criteria is also mentioned. Using the method of Schur complements, the equivalence between positive innerwise and positive definite stability criteria is obtained. This leads to the connection between Sylvester’s resultants and Bezoutians. These formulations form the basis for obtaining all the stability criteria.
From the Hermite criterion for real polynomials, its reduced form — which in fact is the Liénard-Chipart (1918) criterion — is obtained, The connection between Hermite criterion and Lyapunov stability conditions is also indicated thus forging the link between the four stability criteria of Hermite, Routh, Lyapunov, and Hurwitz. Finally, five important applications of the Hurwitz criterion are mentioned.
KeywordsStability Criterion Robust Stability Historical Review Hermite Matrix Principal Minor
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