Remarks on the stability of spatially distributed systems with a cyclic interconnection structure

Mihailo Jovanovic, Murat Arcak, Eduardo D. Sontag

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations


A class of distributed systems with a cyclic interconnection structure is considered. These systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns. In this paper, a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified. For these systems global stability results hold if the "secant" criterion is satisfied. In the linear case, it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic Lyapunov function, which extends a recent diagonal stability result to partial differential equations. For reaction-diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established. All of the derived results remain true for both linear and nonlinear positive diffusion terms. Similar results are shown for compartmental systems.

Original languageEnglish (US)
Title of host publicationProceedings of the 2007 American Control Conference, ACC
Number of pages6
StatePublished - 2007
Event2007 American Control Conference, ACC - New York, NY, United States
Duration: Jul 9 2007Jul 13 2007

Publication series

NameProceedings of the American Control Conference
ISSN (Print)0743-1619


Other2007 American Control Conference, ACC
Country/TerritoryUnited States
CityNew York, NY


  • Biochemical reactions
  • Cyclic interconnections
  • Passivity
  • Secant criterion
  • Spatially distributed systems


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