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Publications of Eduardo D. Sontag jointly with M. Arcak
Articles in journal or book chapters
  1. Z. Aminzare, Y. Shafi, M. Arcak, and E.D. Sontag. Guaranteeing spatial uniformity in reaction-diffusion systems using weighted $L_2$-norm contractions. In V. Kulkarni, G.-B. Stan, and K. Raman, editors, A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, pages 73-101. Springer-Verlag, 2014. [PDF] Keyword(s): contractions, contractive systems, Turing instabilities, diffusion, partial differential equations, synchronization.
    Abstract:
    This paper gives conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations, stated in terms of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain, and similar conditions for diffusively-coupled networks of ordinary differential equations. Also derived are numerical tests making use of linear matrix inequalities that are useful in certifying these conditions.


  2. A. Rufino Ferreira, M. Arcak, and E.D. Sontag. Stability certification of large scale stochastic systems using dissipativity of subsystems. Automatica, 48:2956-2964, 2012. [PDF] Keyword(s): stochastic systems, passivity, noise-to-state stability.
    Abstract:
    This paper deals with the stability of interconnections of nonlinear stochastic systems, using concepts of passivity and noise-to-state stability.


  3. L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with applications to biochemical networks: an input-output approach. IEEE Transactions Autom. Control, 55:1367-1379, 2010. [PDF]
    Abstract:
    This paper provides synchronization conditions for networks of nonlinear systems, where each component of the network itself consists of subsystems represented as operators in the extended L2 space. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of network. The paper also explores results for state-space models as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling.


  4. 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, biochemical 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.


  5. 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): systems biology, biochemical 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.


  6. 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, biochemical 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.


  7. 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, biochemical 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.


  8. L. Moreau, E.D. Sontag, and M. Arcak. Feedback tuning of bifurcations. Systems Control Lett., 50(3):229-239, 2003. [PDF] Keyword(s): bifurcations, adaptive control.
    Abstract:
    This paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.


  9. M. Arcak, D. Angeli, and E.D. Sontag. A unifying integral ISS framework for stability of nonlinear cascades. SIAM J. Control Optim., 40(6):1888-1904, 2002. [PDF] [doi:http://dx.doi.org/10.1137/S0363012901387987] Keyword(s): input to state stability.
    Abstract:
    We analyze nonlinear cascades in which the driven subsystem is integral ISS, and characterize the admissible integral ISS gains for stability. This characterization makes use of the convergence speed of the driving subsystem, and allows a larger class of gain functions when the convergence is faster. We show that our integral ISS gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem.


Conference articles
  1. Y. Shafi, Z. Aminzare, M. Arcak, and E.D. Sontag. Spatial uniformity in diffusively-coupled systems using weighted L2 norm contractions. In Proc. American Control Conference, pages 5639-5644, 2013. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, Turing instabilities, diffusion, partial differential equations, synchronization.
    Abstract:
    We present conditions that guarantee spatial uniformity in diffusively-coupled systems. Diffusive coupling is a ubiquitous form of local interaction, arising in diverse areas including multiagent coordination and pattern formation in biochemical networks. The conditions we derive make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators, and generalize and unify existing theory about asymptotic convergence of trajectories of reaction-diffusion partial differential equations as well as compartmental ordinary differential equations. We present numerical tests making use of linear matrix inequalities that may be used to certify these conditions. We discuss an example pertaining to electromechanical oscillators. The paper's main contributions are unified verifiable relaxed conditions that guarantee synchrony.


  2. A. Rufino Ferreira, M. Arcak, and E.D. Sontag. A decomposition-based approach to stability analysis of large-scale stochastic systems. In Proceedings of the 2012 American Control Conference, Montreal, June 2012, pages Paper FrC10.4, 2012. Keyword(s): stochastic systems, passivity, noise-to-state stability.
    Abstract:
    Conference version of ``Stability certification of large scale stochastic systems using dissipativity of subsystems''.


  3. 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, biochemical networks, systems biology.
    Abstract:
    See abstract and link to pdf in entry for Journal paper.


  4. 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, biochemical networks, systems biology.
    Abstract:
    See abstract and link to pdf in entry for Journal paper.


  5. 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, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.


  6. 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, biochemical 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.


  7. 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, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.


  8. L. Moreau, E.D. Sontag, and M. Arcak. How feedback can tune a bifurcation parameter towards its unknown critical bifurcation value. In Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pages 2401-2406, 2003.


  9. M. Arcak, D. Angeli, and E.D. Sontag. Stabilization of cascades using integral input-to-state stability. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2001, IEEE Publications, 2001, pages 3814-3819, 2001. Keyword(s): nonlinear control, feedback stabilization, input to state stability.



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Last modified: Thu Nov 23 10:40:56 2017
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