The generalized spectral-radius theorem: An analytic-geometric proof

Elsner, Ludwig

doi:10.1016/0024-3795(93)00320-y

Cited by 118 publications

(99 citation statements)

References 4 publications

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“…In discrete-time this was first shown by Berger and Wang [12] and alternative ways of proving the result have been presented by Elsner [47] and Shih, Wu and Pang [151]. On the other hand the result has also been implicit in the Russian literature.…”

Section: Exponential Growth Ratesmentioning

confidence: 84%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

900

558

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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

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Section: Exponential Growth Ratesmentioning

confidence: 84%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

900

558

View full text Add to dashboard Cite

show abstract

“…The main result here (Theorem 6.10) states if SG(M) is not bounded in this case then M has a hyperinvariant subspace. This gives a possibility in Section 7 to prove for precompact sets of compact operators the Berger Wang formula \(M)=r(M ) which was proved in [6] (see also [14]) for operators on finite-dimensional spaces. To realize the convenience of this result, notice that it immediately implies that Volterra semigroups have invariant subspaces.…”

Section: Introductionmentioning

confidence: 96%

Joint Spectral Radius, Operator Semigroups, and a Problem of W. Wojtyński

Shulman

Turovskiı

2000

Journal of Functional Analysis

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We investigate the connections between the invariant subspace problem for operator semigroups and the joint spectral radius. As a consequence, it is proved that any quasinilpotent Lie algebra of compact operators on a Banach space is triangularizable. We extend the Berger Wang formula to precompact sets of essentially scalar operators and prove the continuity of the joint spectral radius on them. Academic Press

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“…Introduction. The joint spectral radius has drawn much attention lately, see, for example, [1,2,3,4,5,6,7,9,13,18] and the references therein, due to its applications in various areas such as switched systems [12], differential equations [8], coding theory [14], and wavelets [15]. For more background material, we also refer the reader to the monograph [11].…”

mentioning

confidence: 99%

On the trace characterization of the joint spectral radius

Xu¹

2010

ELA

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Abstract. A characterization of the joint spectral radius, due to Chen and Zhou, relies on the limit superior of the k-th root of the dominant trace over products of matrices of length k. In this note, a sufficient condition is given such that the limit superior takes the form of a limit. This result is useful while estimating the joint spectral radius. Although it applies to a restricted class of matrices, it appears to be relevant to many realistic situations.

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The generalized spectral-radius theorem: An analytic-geometric proof

Cited by 118 publications

References 4 publications

Stability Criteria for Switched and Hybrid Systems

Stability Criteria for Switched and Hybrid Systems

Joint Spectral Radius, Operator Semigroups, and a Problem of W. Wojtyński

On the trace characterization of the joint spectral radius

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