Control of kinetic plasma instabilities by laser fields

We study the possibility of controlling kinetic plasma instabilities by the reaction of the plasma to applied external electromagnetic fields that could be produced by lasers. We derive the dispersion relation for the corresponding mathematical description, a reduced Vlasov--Maxwell system, by extending the well-known results for the Vlasov--Poisson. It is observed that, under very mild assumptions, the dispersion relation decouples into two parts. The first part is identical to the classic Penrose condition for the Vlasov--Poisson system, while the second part describes the laser field influence on Weibel-type plasma instabilities. In particular, this means that e.g.~a two-stream instability can not be stabilized in this manner as far as linear theory is concerned. We show, however, that nonlinear effects can be used to couple the two parts and achieve effective control. The parameters of the laser (i.e.~the form of the external electric and magnetic fields) are determined by a PDE-constrained optimization approach.