  Plasma instabilities are a major concern in plasma science, for applications
ranging from particle accelerators to nuclear fusion reactors. In this work, we
consider the possibility of controlling such instabilities by adding an
external electric field to the Vlasov--Poisson equations. Our approach to
determining the external electric field is based on conducting a linear
analysis of the resulting equations. We show that it is possible to select
external electric fields that completely suppress the plasma instabilities
present in the system when the equilibrium distribution and the perturbation
are known. In fact, the proposed strategy returns the plasma to its equilibrium
with a rate that is faster than exponential in time. We further perform
numerical simulations of the nonlinear two-stream and bump-on-tail
instabilities to verify our theory and to compare the different strategies that
we propose in this work.
