Convex invertible cones and the Lyapunov equation

Cohen, Nir; Lewkowicz, Izchak

doi:10.1016/0024-3795(95)00424-6

Cited by 46 publications

(36 citation statements)

References 14 publications

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“…where X ⊂ C n×n is a matrix family, providing some properties of sets of matrices possessing common Lyapunov solutions (see Section 3 in [4]). It follows from their observation H X * = (H X ) −1 (see Eq.…”

Section: A Common Lyapunov Solution Formentioning

confidence: 99%

“…It follows from their observation H X * = (H X ) −1 (see Eq. (3.6) in [4]) that if some matrix A has a Lyapunov solution being an involution (recall that a matrix is an involution if it is its own inverse), then that involution is also a Lyapunov solution for A * . The next theorem answers the question when such an involution exists.…”

Section: A Common Lyapunov Solution Formentioning

confidence: 99%

See 1 more Smart Citation

On the existence of a common solution to the Lyapunov equations

Białas¹,

Góra²

2015

Bulletin of the Polish Academy of Sciences Technical Sciences

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Section: A Common Lyapunov Solution Formentioning

confidence: 99%

Section: A Common Lyapunov Solution Formentioning

confidence: 99%

On the existence of a common solution to the Lyapunov equations

Białas¹,

Góra²

2015

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…, Σ Am have a CQLF if and only if the intersection (which is also a convex cone) P {A1,...,Am} = P A1 ∩ P A2 · · · ∩ P Am is non-empty. A number of authors have studied the structure of cones of the form P {A1,...,Am} and how it relates to the problem of CQLF existence [5,6,9,14]. In this context we should also note the work done on the closely related class of cones given by…”

Section: X(t) = A(t)x(t) A(t)mentioning

confidence: 99%

The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem

Mason¹,

Shorten²

2005

ELA

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Abstract. In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence.

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“…Many such methods are based on convex optimization and solution of matrix inequalities, exploiting the fact that the set of quadratic Lyapunov functions for a given linear system is a convex cone. The structure of convex cones of matrices that are closed under matrix inversion and their connections with the algebraic Lyapunov equation is studied in [3]. In [11], the authors consider convex cones associated with quadratic Lyapunov functions for hybrid linear systems.…”

Section: Introductionmentioning

confidence: 99%