Lp stability and delay robustness of network flow control

This paper studies robustness of Kelly´s source and link control laws with respect to disturbances and time-delays. This problem is of practical importance because of unmodeled flows, and propagation and queueing delays, which are ubiquitous in networks. We first show L_p-stability, for p ∈ [l,∞], with respect to additive disturbances. We pursue L_∞-stability within the input-to-state stability (ISS) framework of Sontag which makes explicit the vanishing effect of initial conditions. Next, using this ISS property and a loop transformation, we prove that global asymptotic stability is preserved for sufficiently small time-delays in forward and return channels. For larger delays, we achieve global asymptotic stability by scaling down the control gains as in Paganini et al.