  This paper proposes a robust beamforming (BF) scheme to enhance physical
layer security (PLS) of the downlink of a multibeam satellite system in the
presence of either uncoordinated or coordinated eavesdroppers (Eves).
Specifically, with knowing only the approximate locations of the Eves, we aim
at maximizing the worst-case achievable secrecy rate (ASR) of the legitimate
user (LU), subject to the constraints of per-antenna transmit power and quality
of service (QoS) requirement of the LU. Since the optimization problem is
non-convex, we first adopt the discretization method to deal with the unknown
regions of the Eves and then exploit the log-sum-exp function to approximate
the objective function. Afterwards, a BF method joint alternating direction
method of multipliers (ADMM) with Dinkelbach iteration is presented to solve
this non-convex problem. Finally, simulation results verify that our robust BF
algorithm can effectively improve the security of multibeam satellite systems.
