The chaotic state of the classical Pierce diode is studied using a particle-in-cell computer simulation. The dynamics of the chaotic state is analyzed by estimating ergodic measures of the reconstructed attractor that refine and extend previously reported results [Matsumoto et al., Phys. Plasmas 3 (1996) 177]. The chaos control algorithm suggested by Ott, Grebogi, and Yorke is systematically applied to stabilize periodic orbits up to periodicity four. This demonstrates successful chaos control in a plasma, i.e., a many particle system with long range collective Coulomb interaction.