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.

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.

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