We investigate absolute limits on heat transport in a truncated model of Rayleigh–Bénard convection. Two complementary mathematical approaches–a background method analysis and an optimal control formulation–are used to derive upper bounds in a distinguished eight-ODE model proposed by Gluhovsky, Tong, and Agee. In the optimal control approach the flow no longer obeys an equation of motion, but is instead a control variable. Both methods produce the same estimate, but in contrast to the analogous result for the seminal three-ODE Lorenz system, the best upper bound apparently does not always correspond to an exact solution of the equations of motion.