Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular

For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.

[1]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[2]  Arjan van der Schaft,et al.  On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics , 2011, SIAM J. Appl. Math..

[3]  M. Feinberg.  The existence and uniqueness of steady states for a class of chemical reaction networks , 1995 .

[4]  F. Blanchini.  Constrained control for uncertain linear systems , 1991 .

[5]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: I. the Injectivity Property * , 2006 .

[6]  P. Olver.  Nonlinear Systems , 2013 .

[7]  Olaf Wolkenhauer,et al.  Principal difference between stability and structural stability (robustness) as used in systems biology. , 2007, Nonlinear dynamics, psychology, and life sciences.

[8]  R. Jackson,et al.  General mass action kinetics , 1972 .

[9]  Gaetano Zampieri,et al.  Diffeomorphisms with Banach space domains , 1992 .

[10]  Franco Blanchini,et al.  Structurally robust biological networks , 2011, BMC Systems Biology.

[11]  Franco Blanchini,et al.  Computing the structural influence matrix for biological systems , 2016, Journal of mathematical biology.

[12]  Y. Pyatnitskiy,et al.  Criteria of asymptotic stability of differential and difference inclusions encountered in control theory , 1989 .

[13]  Franco Blanchini,et al.  Structural stability of biochemical networks: Quadratic vs. polyhedral Lyapunov functions , 2015 .

[14]  Martin Feinberg,et al.  Multiple Equilibria in Complex Chemical Reaction Networks: Ii. the Species-reactions Graph , 2022 .

[15]  David Angeli,et al.  Piecewise Linear in rates Lyapunov functions for Complex Reaction Networks , 2013, 52nd IEEE Conference on Decision and Control.

[16]  D. Vecchio,et al.  Biomolecular Feedback Systems , 2014 .

[17]  Roman Srzednicki,et al.  On rest points of dynamical systems , 1985 .

[18]  M. Chaves,et al.  Stability of Rate-Controlled Zero-Deficiency Networks , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  David Richeson,et al.  Addendum to: ``A fixed point theorem for bounded dynamical systems'' [Illinois J. Math. 46 (2002), no. 2, 491--495] , 2004 .

[20]  F. Horn,et al.  On a connexion between stability and graphs in chemical kinetics II. Stability and the complex graph , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  Eduardo D. Sontag.  Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction , 2001, IEEE Trans. Autom. Control..

[22]  Franco Blanchini,et al.  Polyhedral Lyapunov functions for structural stability of biochemical systems in concentration and reaction coordinates , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[23]  David Richeson,et al.  A fixed point theorem for bounded dynamical systems , 2001 .

[24]  David Angeli,et al.  New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions , 2014, IEEE Transactions on Automatic Control.

[25]  David Angeli,et al.  Robust Lyapunov functions for Complex Reaction Networks: An uncertain system framework , 2014, 53rd IEEE Conference on Decision and Control.

[26]  D. Koshland,et al.  An amplified sensitivity arising from covalent modification in biological systems. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Katalin M. Hangos.  Engineering Model Reduction and Entropy-based Lyapunov Functions in Chemical Reaction Kinetics , 2010, Entropy.

[28]  Martin Feinberg,et al.  Multiple steady states for chemical reaction networks of deficiency one , 1995 .

[29]  David F. Anderson,et al.  Global Asymptotic Stability for a Class of Nonlinear Chemical Equations , 2007, SIAM J. Appl. Math..

[30]  U. Alon.  An introduction to systems biology : design principles of biological circuits , 2019 .

[31]  Franco Blanchini,et al.  Piecewise-linear Lyapunov functions for structural stability of biochemical networks , 2014, Autom..

[32]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[33]  F. HoRNt,et al.  On a connexion between stability and graphs in chemical kinetics. I. Stability and the reaction diagram , 1973 .