1. M. Arcak and E.D. Sontag. Passivity-based Stability of Interconnection Structures. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, volume Volume 371, pages 195-204. Springer-Verlag, NY, 2008. [PDF] [doi:10.1007/978-1-84800-155-8_14] Keyword(s): passive systems, secant condition, reaction networks. Abstract:

   In this expository paper, we provide a streamlined version of the key lemma on stability of interconnections due to Vidyasagar and Moylan and Hill, and then show how it its hypotheses may be verified for network structures of great interest in biology.




2. M. Arcak and E.D. Sontag. A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks. Mathematical Biosciences and Engineering, 5:1-19, 2008. Note: Also, preprint: arxiv0705.3188v1 [q-bio], May 2007. [PDF] Keyword(s): MAPK cascades, systems biology, reaction networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems. Abstract:

   This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.




3. M.R. Jovanovic, M. Arcak, and E.D. Sontag. A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:75-86, 2008. Note: Preprint: also arXiv math.OC/0701622, 22 January 2007.[PDF] Keyword(s): MAPK cascades, systems biology, reaction networks, nonlinear stability, nonlinear dynamics, diffusion, secant condition, cyclic feedback systems. 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.




4. M. Arcak and E.D. Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42:1531-1537, 2006. [PDF] Keyword(s): passive systems, systems biology, reaction networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems. Abstract:

   This paper considers a class of systems with a cyclic structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.




5. E.D. Sontag. Passivity gains and the ``secant condition'' for stability. Systems Control Lett., 55(3):177-183, 2006. [PDF] Keyword(s): cyclic feedback systems, systems biology, reaction networks, nonlinear stability, dynamical systems, passive systems, secant condition, reaction networks. Abstract:

   A generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems. The key is the introduction of a quantity that combines gain and phase information for each system in the cascade. For linear one-dimensional systems, the known result is recovered exactly.

1. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with an input-output approach. Part I: Main results. In Proc. IEEE Conf. Decision and Control, Shanhai, Dec. 2009, pages 609-614, 2009. Note: First part of conference version of journal paper.Keyword(s): passive systems, secant condition, reaction networks, systems biology. Abstract:

   See abstract and link to pdf in entry for Journal paper.




2. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with an input-output approach. Part II: State-Space result and application to biochemical networks. In Proc. IEEE Conf. Decision and Control, Shanhai, Dec. 2009, pages 615-620, 2009. Note: Second part of conference version of journal paper.Keyword(s): passive systems, secant condition, reaction networks, systems biology. Abstract:

   See abstract and link to pdf in entry for Journal paper.




3. M. Arcak and E.D. Sontag. A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 2007, pages 4477-4482, 2007. Note: Conference version of journal paper with same title. Keyword(s): systems biology, reaction networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.



4. M.R. Jovanovic, M. Arcak, and E.D. Sontag. Remarks on the stability of spatially distributed systems with a cyclic interconnection structure. In Proceedings American Control Conf., New York, July 2007, pages 2696-2701, 2007. Keyword(s): systems biology, reaction networks, cyclic feedback systems, spatially distributed systems, secant condition. Abstract:

   For distributed systems with a cyclic interconnection structure, a global stability result is shown to hold if the secant criterion is satisfied.




5. M. Arcak and E.D. Sontag. Connections between diagonal stability and the secant condition for cyclic systems. In Proc. American Control Conference, Minneapolis, June 2006, pages 1493-1498, 2006. Keyword(s): systems biology, reaction networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.



6. E.D. Sontag. A notion of passivity gain and a generalization of the `secant condition' for stability. In Proc. IEEE Conf. Decision and Control, Seville, Dec. 2005, IEEE Publications, pages 5645-5649, 2005. Keyword(s): nonlinear stability, dynamical systems.

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