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Abstract

We study the possibility of controlling kinetic plasma instabilities by using lasers to apply external electromagnetic fields. We derive the dispersion relation for the corresponding mathematical description, a reduced Vlasov--Maxwell system, by extending the well-known Penrose condition. 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 influence of the laser on the transverse dynamics (e.g. a Weibel instability). In particular, this means that the longitudinal dynamics (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. This is done by determining the control parameters (i.e. the form of the external electric and magnetic fields) by solving a PDE-constrained optimization problem.