This study investigates relay control for simultaneous wireless information and power transfer in full-duplex relay networks under Nakagami-m fading channels. Unlike previous work, harvest-transmit (HT) and general harvest-transmit-store (HTS) models are respectively considered to maximize average throughput subject to quality of service (QoS) constraints. The end-to-end outage probability of the network in an HT model is presented in an exact integral-form. To prevent outage performance degradation in an HT model, time switching (TS) is designed to maximize average throughput subject to QoS constraints of minimizing outage probability and maintaining a target outage probability, respectively. The optimal TS factors subject to QoS constraints are presented for an HT model. In general, in an HTS model, energy scheduling is performed across different transmission blocks and TS is performed within each block. Compared with the block-based HTS model without TS, the proposed general HTS model can greatly improve outage performance via greedy search (GS). By modeling the relay's energy levels as a Markov chain with a two-stage state transition, the outage probability for GS implementation of the general HTS model is derived. To demonstrate the practical significance of QoS-constrained relay control, numerical results are presented showing that the proposed relay control achieves substantial improvement of outage performance and successful rate.