Controllability analysis for current profile control in tokamaks

The control of the toroidal current density spatial profile in tokamak plasmas will be absolutely critical in future commercial-grade reactors to enable high fusion gain, noninductive sustainment of the plasma current for steady-state operation, and magnetohydrodynamic (MHD) instability-free performance. Since the actuators that are available to achieve a predefined desired current profile are constrained by physical limitations, experiments have shown that some target profiles may not be achievable for all arbitrary initial conditions. This clearly defines a controllability problem, where the transport dynamics of the toroidal current density is governed by a parabolic partial differential equation with diffusivity-interior-boundary actuation. We first prove that the system is not completely controllable and later provide an estimate of the unreachable region.