Stability of a General Class of Distributed Algorithms for Power Control in Time-Varying Wireless Networks

In order for a wireless network to function effectively, the signal power of each user´s transmitter must be sufficiently large to ensure a reliable uplink connection to the receiver, but not so large as to cause interference with neighboring users. We consider a general class of distributed algorithms for the control of transmitter power allocations in wireless networks with a general form of interference nonlinearity. In particular, we allow this interference to have explicit time-dependence, allowing our analysis to remain valid for network configurations that vary with time. We employ appropriately constructed Lyapunov functions to show that any bounded power distribution obtained from these algorithms is uniformly asymptotically stable. Further, we use Lyapunov-Razumikhin functions to show that, even when the system incorporates heterogeneous, time-varying delays, any solution along which the generalized system nonlinearity is bounded must also be uniformly asymptotically stable. Moreover, in both of these cases this stability is shown to be global, meaning that every power distribution has the same asymptotic behavior. These results are also used in the paper to derive time-invariant asymptotic bounds for the trajectories when the system nonlinearities are appropriately bounded.