Optimal Control of a Hot Plasma

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma is located in a bounded container $Ω\subset\mathbb R^3$, for example a fusion reactor. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We model objects, that are placed in space, via given matrix-valued functions $\varepsilon$ (the permittivity) and $μ$ (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting $\partialΩ$ as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimizing problem with a PDE constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions.

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