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We investigate a novel robust flow control framework for heterogeneous network access by devices with multihoming capabilities. Toward this end, we develop an Hinfin-optimal control formulation for allocating rates to devices on multiple access networks with heterogeneous time-varying characteristics. Hinfin analysis and design allow for the coupling between different devices to be relaxed by treating the dynamics for each device as independent of the others. Thus, the distributed end-to-end rate control scheme proposed in this work relies on minimum information and achieves fair and robust rate allocation for the devices. An efficient utilization of the access networks is established through an equilibrium analysis in the static case. We perform measurement tests to collect traces of the available bandwidth on various WLANs and Ethernet. Through simulations, our approach is compared with AIMD and LQG schemes. In addition, the efficiency, fairness, and robustness of the Hinfin-optimal rate controller developed are demonstrated via simulations using the measured real-world network characteristics. Its favorable characteristics and general nature indicate applicability of this framework to a variety of networked systems for flow control.