We investigate the onset of a non-equilibrium phase transition in a one-dimensional ring, constituted by two urns connected by two strands, called active and passive channels. A set of N particles move inside the ring with constant individual speeds; collisions against the channel entries produce reflections with certain probabilities, that differ between active and passive channels. The microscopic dynamics differs from a classical 1D billiard owing to the presence of an interaction mechanism acting inside the active channel, which potentially reverses velocities of its particles. We outline a general theory for the feedback-controlled system which describes quantitatively the phase diagram of the model, based on a mixing property, that is analytically predicted and numerically verified. The probability distributions we define and evolve in time are 1D projections of uniform distributions on d-dimensional spherical surfaces, with d≥1 and d=∞. Consequently results that apply to higher dimensional systems are recovered.