1. M.A. Al-Radhawi, D. Angeli, and E.D. Sontag. On structural contraction of biological interaction networks. 2024. Note: To be submitted. Preprint in: arXiv https://doi.org/10.48550/arXiv.2307.13678.Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms. Abstract:

   In previous work, we have developed an approach to understanding the long-term dynamics of classes of chemical reaction networks, based on rate-dependent Lyapunov functions. In this paper, we show that stronger notions of convergence can be established by proving contraction with respect to non-standard norms. This enables us to show that such networks entrain to periodic inputs. We illustrate our theory with examples from signaling pathways and genetic circuits.




2. 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.




3. Z. Aminzare and E.D. Sontag. Synchronization of diffusively-connected nonlinear systems: results based on contractions with respect to general norms. IEEE Transactions on Network Science and Engineering, 1(2):91-106, 2014. [PDF] Keyword(s): matrix measures, logarithmic norms, synchronization, consensus, contractions, contractive systems. Abstract:

   Contraction theory provides an elegant way to analyze the behavior of certain nonlinear dynamical systems. In this paper, we discuss the application of contraction to synchronization of diffusively interconnected components described by nonlinear differential equations. 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. We base our approach on contraction theory, using matrix measures derived from norms that are not induced by inner products. Such norms are the most appropriate in many applications, but proofs cannot rely upon Lyapunov-like linear matrix inequalities, and different techniques, such as the use of the Perron-Frobenious Theorem in the cases of L1 or L-infinity norms, must be introduced.




4. Z. Aminzare and E.D. Sontag. Logarithmic Lipschitz norms and diffusion-induced instability. Nonlinear Analysis: Theory, Methods & Applications, 83:31-49, 2013. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, Turing instabilities, diffusion, partial differential equations, synchronization. Abstract:

   This paper proves that ordinary differential equation systems that are contractive with respect to $L^p$ norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi-inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.




5. G. Russo, M. di Bernardo, and E.D. Sontag. A contraction approach to the hierarchical analysis and design of networked systems. IEEE Transactions Autom. Control, 58:1328-1331, 2013. [PDF] Keyword(s): contractions, contractive systems, matrix measures, logarithmic norms, synchronization, systems biology. Abstract:

   This paper studies networks of components, and shows that a contraction property on the interconnection matrix, coupled with contractivity of the individual component subsystems, suffices to insure contractivity of the overall system.

1. 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.




2. 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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