Backstepping control of density and energy profiles in a burning tokamak plasma

The control of plasma density profiles is one of the most fundamental problems in fusion reactors. During reactor operation, the spatial profiles of deuterium-tritium fuel, alpha-particles generated by fusion reactions, and energy must be precisely regulated. Here we apply a backstepping boundary control technique to stabilize an unstable equilibrium in a burning plasma. A one-dimensional approximation of the transport equation for energy as well as the density of deuterium-tritium fuel ions and alpha-particles is represented in cylindrical coordinates by a system of partial differential equations (PDEs). To control the ion and energy density profiles, the PDE system is discretized in space using a finite difference method and a backstepping design is applied to obtain a discrete transformation from the original system into an asymptotically stable target system. Numerical simulations of the resulting boundary control law show that the profiles can be successfully controlled with just one step of backstepping.