Spatially Controlled Relay Beamforming: $2$-Stage Optimal Policies

The problem of enhancing Quality-of-Service (QoS) in power constrained, mobile relay beamforming networks, by optimally and dynamically controlling the motion of the relaying nodes, is considered, in a dynamic channel environment. We assume a time slotted system, where the relays update their positions before the beginning of each time slot. Modeling the wireless channel as a Gaussian spatiotemporal stochastic field, we propose a novel $2$-stage stochastic programming problem formulation for optimally specifying the positions of the relays at each time slot, such that the expected QoS of the network is maximized, based on causal Channel State Information (CSI) and under a total relay transmit power budget. This results in a schema where, at each time slot, the relays, apart from optimally beamforming to the destination, also optimally, predictively decide their positions at the next time slot, based on causally accumulated experience. Exploiting either the Method of Statistical Differentials, or the multidimensional Gauss-Hermite Quadrature Rule, the stochastic program considered is shown to be approximately equivalent to a set of simple subproblems, which are solved in a distributed fashion, one at each relay. Optimality and performance of the proposed spatially controlled system are also effectively assessed, under a rigorous technical framework; strict optimality is rigorously demonstrated via the development of a version of the Fundamental Lemma of Stochastic Control, and, performance-wise, it is shown that, quite interestingly, the optimal average network QoS exhibits an increasing trend across time slots, despite our myopic problem formulation. Numerical simulations are presented, experimentally corroborating the success of the proposed approach and the validity of our theoretical predictions.