TY - JOUR
T1 - Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular
AU - Blanchini, Franco
AU - Giordano, Giulia
N1 - Accepted Author Manuscript
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Chemical reaction networks
KW - Dynamical networks
KW - Global stability
KW - Piecewise-linear Lyapunov functions
KW - Structural stability
UR - http://resolver.tudelft.nl/uuid:54ccffe9-9a2e-4608-a632-76d3c74d26f2
UR - http://www.scopus.com/inward/record.url?scp=85029601564&partnerID=8YFLogxK
U2 - 10.1016/j.automatica.2017.08.022
DO - 10.1016/j.automatica.2017.08.022
M3 - Article
AN - SCOPUS:85029601564
SN - 0005-1098
VL - 86
SP - 183
EP - 191
JO - Automatica
JF - Automatica
ER -