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

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