Cross-Positive Matrices

Schneider, Hans; Vidyasagar, M.

doi:10.1137/0707041

Cited by 171 publications

(84 citation statements)

References 5 publications

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“…We note in passing that Lemma 4.5 holds more generally for every polyhedral cone K; see [13,14]. [10] and briefly described in the following.…”

Section: Points Of Nonnegative Potential In This Section a ∈ Rmentioning

confidence: 99%

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Reachability and Holdability of Nonnegative States

Noutsos¹,

Tsatsomeros²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. Linear differential systemsẋ(t) = Ax(t) (A ∈ R n×n , x 0 = x(0) ∈ R n , t ≥ 0) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of A furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some a ≥ 0, A + aI must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions x 0 that result in nonnegative states x(t) in finite time are shown to form a convex cone that is related to the matrix exponential e tA and its eventual nonnegativity.

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“…We note in passing that Lemma 4.5 holds more generally for every polyhedral cone K; see [13,14]. [10] and briefly described in the following.…”

Section: Points Of Nonnegative Potential In This Section a ∈ Rmentioning

confidence: 99%

“…This is equivalent to asking whether or not e tA K ⊆ K for all t ≥ 0. To resolve this question, we will invoke the following extension of Lemma 3.1 from R n + to simplicial cones, which can be found in [13,14]. …”

Section: Points Of Nonnegative Potential In This Section a ∈ Rmentioning

confidence: 99%

Reachability and Holdability of Nonnegative States

Noutsos¹,

Tsatsomeros²

2008

SIAM J. Matrix Anal. & Appl.

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“…Let λ be the minimum eigenvalue of A. Then a corresponding eigenvector u is in K (see Theorem 6,[17]). It follows that 0 ≤ A(u), u = λ||u|| 2 , so λ ≥ 0.…”

Section: Z Transformationsmentioning

confidence: 99%

Pseudomonotonicity and related properties in Euclidean Jordan algebras

Tao

2011

ELA

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“…The most well known result is the condition of non-negativity on A which states that if A ij ≥ 0 for i = j, then non-negative initial conditions yield non-negative solutions. Schneider and Vidyasagar [33] introduced the notion of cross-positivity of A on Γ n which was shown to be equivalent to exponential non-negativity. Meyer et al [34] extended cross-positivity to nonlinear fields.…”

Section: Theorem 1 (27 In [31])mentioning

confidence: 99%

Cone invariance and rendezvous of multiple agents

Bhattacharya

Tiwari

Fung

et al. 2009

Proceedings of the Institution of Mechanical Engineers, Part G:

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In this paper we present a dynamical systems framework for analyzing multi-agent rendezvous problems and characterize the dynamical behavior of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modeled as linear first order systems. The proposed framework however is not fundamentally limited to linear first order dynamics and can be extended to analyze rendezvous of higher order agents.In the proposed framework, the problem of rendezvous is cast as a stabilization problem, with a set of constraints on the trajectories of the agents defined on the phase plane. We pose the n-agent rendezvous problem as an ellipsoidal cone invariance problem in the n dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented. Index TermsMulti-agent rendezvous, cooperative dynamical systems, monotone systems, cone invariance, non-negative matrices.

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Cross-Positive Matrices

Cited by 171 publications

References 5 publications

Reachability and Holdability of Nonnegative States

Reachability and Holdability of Nonnegative States

Pseudomonotonicity and related properties in Euclidean Jordan algebras

Cone invariance and rendezvous of multiple agents

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