A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure. Jovanovic, M., Arcak, M., & Sontag, E. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:75-86, 2008. Preprint: also arXiv math.OC/0701622, 22 January 2007.abstract bibtex 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.
@ARTICLE{IEEEsysbio_JAS,
AUTHOR = {M.R. Jovanovic and M. Arcak and E.D. Sontag},
JOURNAL = {IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology},
TITLE = {A passivity-based approach to stability of spatially
distributed systems with a cyclic interconnection structure},
YEAR = {2008},
OPTMONTH = {},
NOTE = {Preprint: also arXiv math.OC/0701622, 22 January 2007.},
OPTNUMBER = {},
PAGES = {75-86},
VOLUME = {55},
KEYWORDS = {MAPK cascades, systems biology, biochemical networks,
nonlinear stability, nonlinear dynamics, diffusion, secant condition,
cyclic feedback systems},
PDF = {../../FTPDIR/jovanovic_arcak_sontag_TAC_CAS2008.pdf},
ABSTRACT = {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.}
}
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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.","bibtex":"@ARTICLE{IEEEsysbio_JAS,\n AUTHOR = {M.R. Jovanovic and M. Arcak and E.D. 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In this paper, a class of cyclic systems in \n which addition of diffusion does not have a destabilizing effect is \n identified. For these systems global stability results hold if the \n \"secant\" criterion is satisfied. In the linear case, it is shown that \n the secant condition is necessary and sufficient for the existence of \n a decoupled quadratic Lyapunov function, which extends a recent \n diagonal stability result to partial differential equations. For \n reaction-diffusion equations with nondecreasing coupling \n nonlinearities global asymptotic stability of the origin is \n established. All of the derived results remain true for both linear \n and nonlinear positive diffusion terms. 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