Differential Game for Distributed Power Control in Device-to-Device Communications Underlaying Cellular Networks

In the formulating of power control for wireless networks, the radio channel is commonly formulated using static models of optimization or game theory. In these models, the optimization programming or static game is played from time point to time point. Therefore, this approach neglects the dynamics of the time-varying channels and assumes the statistical independence between the successive time points. In this paper, we utilize differential equations to model the wireless links, then formulate a differential game for the power control problem in device-to-device (D2D) communications underlaying cellular networks. The game players are the D2D pairs, which manage their transmit power by solving the continuous-time optimal control problems. The time-dependant cost function allows us to optimize the long-term expected cost, instead of point-wise instantaneous cost. We formulate the problem in an affine quadratic form that admits analytical solutions. The unique feedback Nash equilibrium of the game is shown to exist. From a stochastic optimal control algorithm, we design a distributed power control mechanism that converges to the game’s equilibrium. The simulation results show that the proposed approach achieves significant performance improvement compared to the point-wise based approaches.