Incremental Stability of Traffic Reaction Models

Licentiatavhandling, 2025

Incremental asymptotic stability is assessed for cooperative systems of ordinary differential equations (ODEs). Such systems of ODEs arise in macroscopic traffic flow modeling which is emphasized in the present thesis. If a system of ODEs is incrementally asymptotically stable, then there exists a set of initial conditions from which all solutions converge to each other asymptotically and this can be exploited in a state estimation context.

It is shown that if the state space of a cooperative system of ODEs is a Cartesian product of intervals, then this system is incrementally asymptotically stable if and only if all solutions that are initially ordered, converge to each other in an appropriate sense. This fact is used to establish incremental asymptotic stability for a class Traffic Reaction Models.

The Traffic Reaction Model forms a family of numerical schemes to solve scalar conservation laws, governed by partial differential equations (PDEs). For one conservation law there are several numerical schemes and if the scheme is semi-discrete it gives rise to a system of ODEs. Suitable conditions on the conservation law are provided such that a particular semi-discrete scheme gives rise to an incrementally exponentially stable system of ODEs.