Stability Analysis of Compartmental and Cooperative Systems
Sondre Wiersdalen, Mike Pereira, Annika Lang, Gabor Szederkenyi, Jean Auriol, Balazs Kulcsar
TL;DR
The paper addresses stability of nonlinear, nonautonomous compartmental systems in which the right-hand side has the form $Q(t,q)=F(t,q)\,g(t,q)+I(t,q)$. It develops a constructive Lyapunov framework by permuting compartmental matrices into an outflow canonical form and proving a main theorem that yields exponential stability of the null solution for all $F$ in a structured family $\mathcal{F}(a,b,l)$, via a linear Lyapunov function $V(q)=\langle v,q\rangle$ with $v>0$ and rate $\lambda>0$. It then treats cooperative systems in a box, establishing necessary and sufficient conditions for incremental asymptotic and exponential stability (IAS/IES) based on order-preserving dynamics and a contractive inequality, and shows how these results apply to a class of discretized conservation laws modeling traffic through an intersection of compartmental and cooperative structures. Traffic Reaction Models are used as concrete illustrations, demonstrating that discretized conservation laws yield incrementally exponentially stable dynamics under suitable flux and boundary conditions. Overall, the work provides practical, checkable stability criteria for mass-conserving networks and connects analytical results to numerical schemes in traffic-flow modeling.
Abstract
The present article considers stability of the solutions to nonlinear and nonautonomous compartmental systems governed by ordinary differential equations (ODEs). In particular, compartmental systems with a right-hand side that can be written as a product of a matrix function and vector function. Sufficient, and on occasion necessary, conditions on the matrix function are provided to conclude exponential stability of the null solution. The conditions involve verifying that the matrix function takes its values in a set of compartmental matrices on a certain canonical form, and are easy to check. Similar conditions are provided to establish incremental exponential stability for compartmental systems governed by cooperative systems of ODEs. The solutions to such systems satisfy a so-called ordering. Systems that are cooperative in a box, are shown to be incrementally asymptotically stable if and only if every pair of initially ordered solutions converge to each other. Traffic Reaction Models are used to illustrate the results, which are numerical schemes to solve conservation laws in one spatial dimension. Suitable conditions on the flux function of the conservation law are given such that the numerical scheme gives rise to an incrementally exponentially stable system.