1. Z. Aminzare and E.D. Sontag. Some remarks on spatial uniformity of solutions of reaction-diffusion PDEs. Nonlinear Analysis, 147:125-144, 2016. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, synchronization, consensus, reaction-diffusion PDEs, partial differential equations. Abstract:

   This paper presents a condition which guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction diffusion partial differential equation (PDE) with Neumann boundary conditions in one dimension, using the Jacobian matrix of the reaction term and the first Dirichlet eigenvalue of the Laplacian operator on the given spatial domain. The estimates are based on logarithmic norms in non-Hilbert spaces, which allow, in particular for a class of examples of interest in biology, tighter estimates than other previously proposed methods.

1. Z. Aminzare and E.D. Sontag. Contraction methods for nonlinear systems: A brief introduction and some open problems. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pages 3835-3847, 2014. [PDF] Keyword(s): contractions, contractive systems, stability, reaction-diffusion PDE's, synchronization, contractive systems, stability. Abstract:

   Contraction theory provides an elegant way to analyze the behaviors of certain nonlinear dynamical systems. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that trajectories converge to each other. The present paper provides a self-contained introduction to some of the basic concepts and results in contraction theory, discusses applications to synchronization and to reaction-diffusion partial differential equations, and poses several open questions.




2. Z. Aminzare and E.D. Sontag. Remarks on diffusive-link synchronization using non-Hilbert logarithmic norms. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pages 6086-6091, 2014. Keyword(s): contractions, contractive systems, stability, reaction-diffusion PDE's, synchronization. Abstract:

   In this paper, we sketch recent results for synchronization in a network of identical ODE models which are diffusively interconnected. In particular, we provide estimates of convergence of the difference in states between components, in the cases of line, complete, and star graphs, and Cartesian products of such graphs.

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