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Publications about 'contractions'
Articles in journal or book chapters
  1. M. Margaliot, E.D. Sontag, and T. Tuller. Checkable conditions for contraction after small transients in time and amplitude. In N. Petit, editor, Feedback Stabilization of Controlled Dynamical Systems - In Honor of Laurent Praly, volume 473 of Lecture Notes in Control and Inform. Sci., pages 279-305. Springer-Verlag, Berlin, 2017. [PDF] Keyword(s): contractions, contractive systems, stability.
    Abstract:
    This is an expository paper, which compares in detail various alternative weak contraction ideas for nonlinear system stability.


  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. M. Margaliot, E.D. Sontag, and T. Tuller. Contraction after small transients. Automatica, 67:178-184, 2016. [PDF] Keyword(s): entrainment, nonlinear systems, stability, contractions, contractive systems.
    Abstract:
    Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We introduce three new forms of generalized contraction (GC) that are motivated by allowing contraction to take place after small transients in time and/or amplitude. These forms of GC are useful for several reasons. First, allowing small transients does not destroy the asymptotic properties provided by standard contraction. Second, in some cases as we change the parameters in a contractive system it becomes a GC just before it looses contractivity. In this respect, GC is the analogue of marginal stability in Lyapunov stability theory. We provide checkable sufficient conditions for GC, and demonstrate their usefulness using several models from systems biology that are not contractive, with respect to any norm, yet are GC.


  4. A. Raveh, M. Margaliot, E.D. Sontag, and T. Tuller. A model for competition for ribosomes in the cell. Proc. Royal Society Interface, 13:2015.1062, 2016. [PDF] Keyword(s): resource competition, ribosomes, entrainment, nonlinear systems, stability, contractions, contractive systems.
    Abstract:
    We develop and analyze a general model for large-scale simultaneous mRNA translation and competition for ribosomes. Such models are especially important when dealing with highly expressed genes, as these consume more resources. For our model, we prove that the compound system always converges to a steady-state and that it always entrains or phase locks to periodically time-varying transition rates in any of the mRNA molecules. We use this model to explore the interactions between the various mRNA molecules and ribosomes at steady-state. We show that increasing the length of an mRNA molecule decreases the production rate of all the mRNAs. Increasing any of the codon translation rates in a specific mRNA molecule yields a local effect: an increase in the translation rate of this mRNA, and also a global effect: the translation rates in the other mRNA molecules all increase or all decrease. These results suggest that the effect of codon decoding rates of endogenous and heterologous mRNAs on protein production might be more complicated than previously thought.


  5. 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.


  6. 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.


  7. M. Margaliot, E.D. Sontag, and T. Tuller. Entrainment to periodic initiation and transition rates in a computational model for gene translation. PLoS ONE, 9(5):e96039, 2014. [WWW] [PDF] [doi:10.1371/journal.pone.0096039] Keyword(s): ribosomes, entrainment, nonlinear systems, stability, contractions, contractive systems.
    Abstract:
    A recent biological study has demonstrated that the gene expression pattern entrains to a periodically varying abundance of tRNA molecules. This motivates developing mathematical tools for analyzing entrainment of translation elongation to intra-cellular signals such as tRNAs levels and other factors affecting translation. We consider a recent deterministic mathematical model for translation called the Ribosome Flow Model (RFM). We analyze this model under the assumption that the elongation rate of the tRNA genes and/or the initiation rate are periodic functions with a common period T. We show that the protein synthesis pattern indeed converges to a unique periodic trajectory with period T. The analysis is based on introducing a novel property of dynamical systems, called contraction after a short transient (CAST), that may be of independent interest. We provide a sufficient condition for CAST and use it to prove that the RFM is CAST, and that this implies entrainment. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and suggest a new approach for engineering genes to obtain a desired, periodic, synthesis rate.


  8. 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.


  9. 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.


  10. E.D. Sontag. Contractive systems with inputs. In Jan Willems, Shinji Hara, Yoshito Ohta, and Hisaya Fujioka, editors, Perspectives in Mathematical System Theory, Control, and Signal Processing, pages 217-228. Springer-verlag, 2010. [PDF] Keyword(s): contractions, contractive systems, consensus, synchronization.
    Abstract:
    Contraction theory provides an elegant way of analyzing the behaviors of systems subject to external inputs. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that all trajectories converge to a unique solution. This property is especially interesting when forcing functions are periodic (globally attracting limit cycles result), as well as in the context of establishing synchronization results. The present paper provides a self-contained introduction to some basic results, with a focus on contractions with respect to non-Euclidean metrics.


  11. G. Russo, M. di Bernardo, and E.D. Sontag. Global entrainment of transcriptional systems to periodic inputs. PLoS Computational Biology, 6:e1000739, 2010. [PDF] Keyword(s): contractive systems, contractions, systems biology, biochemical networks, gene and protein networks.
    Abstract:
    This paper addresses the problem of giving conditions for transcriptional systems to be globally entrained to external periodic inputs. By using contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all solutions converge to fixed limit cycles. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems.


Conference articles
  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.


  3. E.D. Sontag, M. Margaliot, and T. Tuller. On three generalizations of contraction. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pages 1539-1544, 2014. Keyword(s): contractions, contractive systems, stability.
    Abstract:
    We introduce three forms of generalized contraction~(GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior. We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are~GC.


  4. 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.


  5. G. Russo, M. di Bernardo, and E.D. Sontag. Stability of networked systems: a multi-scale approach using contraction. In Proc. IEEE Conf. Decision and Control, Atlanta, Dec. 2010, pages FrB14.3, 2010. Keyword(s): contractive systems, contractions, systems biology, biochemical networks, synchronization.
    Abstract:
    Preliminary conference version of ''A contraction approach to the hierarchical analysis and design of networked systems''.



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