# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Diagonal stability of matrices with cyclic structure and the secant condition. (English) Zbl 1159.93022
Summary: An existence result on diagonal solutions of a linear matrix inequality is used to study diagonal Hurwitz and Schur stability and to derive the secant condition for systems with cyclic structure.
##### MSC:
 93D09 Robust stability of control systems 93C15 Control systems governed by ODE 15A39 Linear inequalities of matrices
##### References:
 [1] Tyson, J. J.; Othmer, H. G.: The dynamics of feedback control circuits in biochemical pathways, Progress in theoretical biology, vol. 5 5, 1-62 (1978) · Zbl 0448.92010 [2] Thron, C. D.: The secant condition for instability in biochemical feedback control, I, Bull. math. Biol. 53, 383-401 (1991) · Zbl 0751.92002 [3] Arcak, M.; Sontag, E. D.: Diagonal stability for a class of cyclic systems and its connection with the secant condition, Automatica 42, 1531-1537 (2006) · Zbl 1132.39002 · doi:10.1016/j.automatica.2006.04.009 [4] Jovanović, M. R.; Arcak, M.; Sontag, E. D.: A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure, IEEE trans. Circuits systems 55, 75-86 (2008) [5] Kholodenko, B. N.: Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades, Eur. J. Biochem. 267, 1583-1588 (2001) [6] Sontag, E. D.: Passivity gains and the secant condition for stability, Systems control lett. 55, 177-183 (2006) · Zbl 1129.93476 · doi:10.1016/j.sysconle.2005.06.010 [7] Dashkovskiy, S.; Rüffer, B. S.; Wirth, F. R.: An ISS small gain theorem for general networks, Math. control signals systems 19, 93-122 (2007) · Zbl 1125.93062 · doi:10.1007/s00498-007-0014-8 [8] Arcak, M.; Sontag, E. D.: A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks, Math. biosci. Eng. 5, 1-19 (2008) [9] Hsu, L.; Kaszkurewicz, E.; Bhaya, A.: Matrix-theoretic conditions for the realizability of sliding manifolds, Systems control lett. 40, 145-152 (2000) · Zbl 0977.93018 · doi:10.1016/S0167-6911(00)00013-X [10] Hofbauer, J.; Sigmund, K.: Evolutionary games and population dynamics, (1998) [11] Dussy, S.: Robust diagonal stabilization and finite precision problem: an LMI approach, IEEE trans. Automat. control 45, 125-128 (2000) · Zbl 0972.93060 · doi:10.1109/9.827368 [12] Kaszkurewicz, E.; Bhaya, A.: Matrix diagonal stability in systems and computation, (2000) · Zbl 0951.93058