Persistently damped transport on a network of circles

TL;DR

This paper proves exponential stability for a network of coupled transport equations on circles with intermittent damping, using explicit solution formulas and conditions on the network's parameters.

Contribution

It establishes conditions under which a network of circles with intermittent damping is exponentially stable, extending previous results to more complex network configurations.

Findings

System is exponentially stable under certain conditions on matrix M and circle lengths.

Explicit solution formulas enable tracking effects of intermittent damping.

Stability is uniform with respect to persistently exciting signals.

Abstract

In this paper we address the exponential stability of a system of transport equations with intermittent damping on a network of $N \geq 2$ circles intersecting at a single point $O$. The $N$ equations are coupled through a linear mixing of their values at $O$, described by a matrix $M$. The activity of the intermittent damping is determined by persistently exciting signals, all belonging to a fixed class. The main result is that, under suitable hypotheses on $M$ and on the rationality of the ratios between the lengths of the circles, such a system is exponentially stable, uniformly with respect to the persistently exciting signals. The proof relies on an explicit formula for the solutions of this system, which allows one to track down the effects of the intermittent damping.