Power Control under Finite Power Constraints ∗

1. Introduction. In wireless communication, it is common that multiple users share the same radio spectrum simultaneously. Signal of one channel appears then as noise to the other channels. Hence, users transmitting power at power levels above than what is necessary to maintain a given level of signal-to-noise quality are not only wasteful of their battery power, they also cause unnecessary noise to other users. As a result, other users may try to compensate their signal quality by increasing their transmission power. This racing condition due to mutual interference could lead to an unstable mode in which all users try to raise their power levels without achieving any real gain in the overall signal-to-noise ratio. The objective of the power control problem is to ameliorate these adverse mutual interference effects (see for example [1-3]) as well as effects from fading effects and thermal noises. Generally speaking, the power control problem should be viewed as a distributed, nonlinear, constrained optimal control problem. In certain cases, it can also be regarded as a multiuser game-theoretic problem. Due to the problem complexity, it is common to simplify it to make the problem amendable to analysis. For a classification of the power control problems usually discussed in the literature, we refer to [4]. A crucial structure of the power control problem is the mapping between the power levels and the carrier-to-interfere ratio (essentially, the signal-to-noise ratio). Let G ij represent the channel gain between the i-th receiver and the j-th transmitter. In practice, this gain is an unknown time-varying parameter, depending on the distance between the transmitter and receiver as well as on the fading effects. The matrix G = (G ij) is known as the channel gain matrix. The carrier-to-interference ratio of the i-th receiver is defined as