On Lyapunov boundary control of unstable magnetohydrodynamic plasmas

Abstract

Starting from a simple marginally stable model considered for Lyapunov based boundary control of flexible mechanical systems, we add a term driving an instability and prove that for an appropriate control condition the system can become Lyapunov stable. A similar approximate extension is found for the general energy principle of linearized magnetohydrodynamics. The implementation of such external instantaneous actions may, however, impose challenging constraints for fusion plasmas.

Energy principle for the modes interacting with a resistive wall in toroidal systems

“…Note that with (37) and (53) we actually proved the validity of ( 50) and ( 54) near the idealwall limit without the use of (20). This is an approximation, but a step ahead with respect to the standard model with an ideal wall [1][2][3][4][5][6][7] and its improvements incorporating a resistive wall [25][26][27][28][29][30][31][32][33].…”

“…The final relation (61) is the energy balance that incorporates the sink of the perturbed energy due to the wall resistivity. It differs from the known energy principles with a resistive wall [25][26][27][28][29][30][31][32] by the terms representing the energy loss outside the plasma. We calculated them in a general form under the assumption that s d w with account of the first-order corrections in the expansion in s/d w .…”

“…The model is based on the standard set of linearized magnetohydrodynamic (MHD) equations [1][2][3][4][5][6][7] for the plasma dynamics in the toroidal systems, but here the plasma and the wall of the vacuum chamber are not assumed ideal. As a consequence, the force operator is non-Hermitian, and the resulting torque-energy balance contains a dissipative term due to the wall resistivity, as in general formulas in [25][26][27]. In [28][29][30] this term was calculated in the thin-wall approximation valid for slow RWMs, at s d w with s the skin depth and d w the wall thickness.…”