We study adjustable robust network design for potential-based flows with controllable elements under load uncertainty. The resulting problem combines discrete here-and-now expansion decisions with wait-and-see operational decisions governed by nonconvex flow constraints. Moreover, controllable elements introduce adjustable integer decisions, which are algorithmically challenging. We equivalently characterize robust feasibility and robust optimality of a fixed network design using adversarial bilevel problems. For robust feasibility, we extend an existing characterization for potential-based networks without controllable elements to networks with controllable elements under the structural assumption that no controllable element is part of a cycle. This yields a characterization of robust feasibility consisting of polynomially many mixed-integer nonlinear bilevel problems. Since controllable elements make the objective depend on both here-and-now expansion cost and wait-and-see operating cost, we verify robust optimality of a fixed network design using an additional mixed-integer nonlinear bilevel problem. We then derive equivalent single-level reformulations of these bilevel problems under the stated structural assumption. Building on these reformulations, we present an exact adversarial solution approach for computing adjustable robust network designs with controllable elements and demonstrate its applicability to gas networks. More generally, the developed potential-based framework can be used to compute robust network designs for different types of utility networks with controllable elements, including hydrogen and water networks.