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Feedback stabilization of magnetohydrodynamic (MHD) modes is studied in acylindrical model for a tokamak with resistivity, viscosity and toroidalrotation. The control is based on a linear combination of the normal andtangential components of the magnetic field just inside the resistive wall. Thefeedback includes complex gain, for both the normal and for the tangentialcomponents, and the imaginary part of the feedback for the former is equivalentto plasma rotation. The work includes (1) analysis with a reduced resistive MHDmodel for a tokamak with finite \beta and with stepfunction current density andpressure profiles, and (2) computations with full compressible visco-resistiveMHD and smooth decreasing profiles of current density and pressure. Theequilibria are stable for \beta=0 and the marginal stability values$\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$ (resistive plasma,resistive wall; resistive plasma, ideal wall; ideal plasma, resistive wall;ideal plasma, ideal wall) are computed for both cases. The main results are:(a) imaginary gain with normal sensors or plasma rotation stabilizes below$\beta_{rp,iw}$ because rotation supresses the diffusion of flux from theplasma out through the wall and, more surprisingly, (b) rotation or imaginarygain with normal sensors destabilizes above $\beta_{rp,iw}$ because it preventsthe feedback flux from entering the plasma through the resistive wall to form avirtual wall. The effect of imaginary gain with tangential sensors is morecomplicated but essentially destabilizes above and below $\beta_{rp,iw}$. Amethod of using complex gain to optimize in the presence of rotation in the$\beta>\beta_{rp,iw}$ regime is presented.

Control of resistive wall modes in a cylindrical tokamak with plasma rotation and complex gain