Geodesic
Verified stability analysis using the Lyapunov matrix equation
Electronic transactions on numerical analysis, Tome 40 (2013), pp. 187-203.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The Lyapunov matrix equation $AX + XA^* = C$ arises in many applications, particularly in the context of stability of matrices or solutions of ordinary differential equations. In this paper we present a method, based on interval arithmetic, which computes with mathematical rigor an interval matrix containing the exact solution of the Lyapunov equation. We work out two options which can be used to verify, again with mathematical certainty, that the exact solution of the equation is positive definite. This allows to prove stability of the (non-Hermitian) matrix $A$ if we chose $C$ as a negative definite Hermitian matrix. Our algorithm has computational cost comparable to that of a state-of-the art algorithm for computing a floating point approximation of the solution because we can cast almost all operations as matrix-matrix operations for which interval arithmetic can be implemented very efficiently.
Classification : 65F05, 65G20
Keywords: stability analysis, Lyapunov matrix equation, interval arithmetic, krawczyk's method, verified computation 
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     author = {Frommer, Andreas and Hashemi, Behnam},
     title = {Verified stability analysis using the {Lyapunov} matrix equation},
     journal = {Electronic transactions on numerical analysis},
     pages = {187--203},
     publisher = {mathdoc},
     volume = {40},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2013__40__a16/}
}
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Frommer, Andreas; Hashemi, Behnam. Verified stability analysis using the Lyapunov matrix equation. Electronic transactions on numerical analysis, Tome 40 (2013), pp. 187-203. http://geodesic.mathdoc.fr/item/ETNA_2013__40__a16/