- ▪Z. Aminzare, E.D. Sontag, "Some remarks on spatial uniformity of solutions of reaction-diffusion PDEs", Nonlinear Analysis, vol. 147, pp. 125-144, 2016. pdfcontractions · 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.
- ▪Z. Aminzare, E.D. Sontag, "Remarks on diffusive-link synchronization using non-Hilbert logarithmic norms", In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pp. 6086-6091, 2014.
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.