Reachability and Holdability of Nonnegative States

Noutsos, D.; Tsatsomeros, Michael J.

doi:10.1137/070693850

Cited by 80 publications

(83 citation statements)

References 10 publications

(17 reference statements)

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“…Other matrices J that yield a positive SSIM M , but are not associated with cooperative systems, can be found based on a quantitative approach: this is the case of eventually nonnegative matrices [21], [22] with a proper diagonal shift.…”

Section: B Quantitative Criteriamentioning

confidence: 99%

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

Giordano

Altafini

2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

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Abstract-In stable biological and ecological networks, the steady-state influence matrix gathers the signs of steady-state responses to step-like perturbations affecting the variables. Such signs are difficult to predict a priori, because they result from a combination of direct effects (deducible from the Jacobian of the network dynamics) and indirect effects. For stable monotone or cooperative networks, the sign pattern of the influence matrix can be qualitatively determined based exclusively on the sign pattern of the system Jacobian. For other classes of networks, we show that a semi-qualitative approach yields sufficient conditions for Jacobians with a given sign pattern to admit a fully positive influence matrix, and we also provide quantitative conditions for Jacobians that are translated eventually nonnegative matrices. We present a computational test to check whether the influence matrix has a constant sign pattern in spite of parameter variations, and we apply this algorithm to quasi-Metzler Jacobian matrices, to assess whether positivity of the influence matrix is preserved in spite of deviations from cooperativity. When the influence matrix is fully positive, we give a simple vertex algorithm to test robust stability. The devised criteria are applied to analyse the steadystate behaviour of ecological and biomolecular networks.

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Section: B Quantitative Criteriamentioning

confidence: 99%

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

Giordano

Altafini

2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

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“…Eventually exponentially positive matrices have applications to dynamical systems in situations where it is of interest to determine whether an initial trajectory reaches positivity at a certain time and remains positive thereafter [Noutsos and Tsatsomeros 2008]. There is a characterization of eventual exponential positivity in terms of eventual positivity:…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.5 was motivated by the following test for eventual exponential positivity, which is implicit in the proof of Theorem 3.3 in [Noutsos and Tsatsomeros 2008] (and also follows immediately from that theorem, which is Theorem 1.1 above). An eventually positive matrix must have a positive entry in each row and column.…”

Section: Introductionmentioning

confidence: 99%

Potentially eventually exponentially positive sign patterns

Archer

Catral

Erickson

et al. 2013

Involve

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We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP sign patterns. A characterization of all 2 × 2 and 3 × 3 PEEP sign patterns is given.

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“…These systems are dynamical systems whose state variables are nonnegative at all times. In [13,18,25], the positive (nonnegative) singular systems have been widely developed.…”

Section: Introductionmentioning

confidence: 99%

An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index

Herrero

Ramı́rez

Thome

2014

Linear and Multilinear Algebra

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This paper deals with singular systems of index k ≥ 1. Our main goal is to find a state-feedback such that the closed-loop system satisfies the regularity condition and it is nonnegative and stable. In order to do that, the core-nilpotent decomposition of a square matrix is applied to the singular matrix of the system. Moreover, if the Drazin projector of this matrix is nonnegative then the previous decomposition allows us to write the core-part of the matrix in a specific block form. In addition, an algorithm to study this kind of systems via a state-feedback is designed.

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Reachability and Holdability of Nonnegative States

Cited by 80 publications

References 10 publications

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

Potentially eventually exponentially positive sign patterns

An algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index

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