Incremental Exponential Stability of the Unidirectional Flow Model

Preprint, 2023

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.