  Fusion energy offers the potential for the generation of clean, safe, and
nearly inexhaustible energy. While notable progress has been made in recent
years, significant challenges persist in achieving net energy gain. Improving
plasma confinement and stability stands as a crucial task in this regard and
requires optimization and control of the plasma system. In this work, we deploy
a PDE-constrained optimization formulation that uses a kinetic description for
plasma dynamics as the constraint. This is to optimize, over all possible
controllable external electric fields, the stability of the plasma dynamics
under the condition that the Vlasov--Poisson (VP) equation is satisfied. For
computing the functional derivative with respect to the external field in the
optimization updates, the adjoint equation is derived. Furthermore, in the
discrete setting, where we employ the semi-Lagrangian method as the forward
solver, we also explicitly formulate the corresponding adjoint solver and the
gradient as the discrete analogy to the adjoint equation and the Frechet
derivative. A distinct feature we observed of this constrained optimization is
the complex landscape of the objective function and the existence of numerous
local minima, largely due to the hyperbolic nature of the VP system. To
overcome this issue, we utilize a gradient-accelerated genetic algorithm,
leveraging the advantages of the genetic algorithm's exploration feature to
cover a broader search of the solution space and the fast local convergence
aided by the gradient information. We show that our algorithm obtains good
electric fields that are able to maintain a prescribed profile in a beam
shaping problem and uses nonlinear effects to suppress plasma instability in a
two-stream configuration.
