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 t...

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Bibliographic Details
Published inAutomatica (Oxford) Vol. 86; pp. 183 - 191
Main Authors Blanchini, Franco, Giordano, Giulia
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.12.2017
Subjects
Chemical reaction networks
Control Engineering
Dynamical networks
Electrical Engineering, Electronic Engineering, Information Engineering
Elektroteknik och elektronik
Engineering and Technology
Global stability
Piecewise-linear Lyapunov functions
Reglerteknik
Structural stability
Teknik
Piecewise-linear Lyapunov functions
Structural stability
Global stability
Dynamical networks
Chemical reaction networks
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Summary: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.
ISSN:0005-1098
1873-2836
DOI:10.1016/j.automatica.2017.08.022