Loss of continuity in cellular networks under stabilizing transmit power control

Recently, passivity-based techniques to control the transmit power of mobile nodes have been proposed to ensure the finite gain stability of a class of cellular CDMA networks. These techniques implement the Zames-Falb multipliers at the mobile node and at the base stations. The finite gain stability of such a cellular network follows as a consequence of the fact that the Zames-Falb multipliers preserve positivity of monotone memoryless nonlinearities. In this note, we show that such a cellular network may not be a continuous system. Hence, the following undesirable scenarios may occur if the set-point is varied with time: (i) the Nash equilibrium may cease to exist, and (ii) vanishingly small changes in set-points (such as the target SINR values) may cause the system trajectory to jump from one equilibrium point to another, leading to nonvanishingly small variations in the mobile transmit powers and the actual SINR values across such networks.